Mixing of asymmetric logarithmic suspension flows over interval exchange transformations.
Corinna Ulcigrai\footnote{Mathematics Department
We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations.
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\title{Mixing of asymmetric logarithmic suspension flows over interval exchange transformations.}

\author{Corinna Ulcigrai\footnote{Mathematics Department, Princeton University.
{\it Email: \,}% address:} {
ulcigrai@math.princeton.edu}}

\date{}

\begin{document}

\maketitle

\begin{abstract}
We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations.
\end{abstract}

\section{Introduction.}
\subsection{Motivation and main references.}
Hamiltonian systems with  multi-valued Hamiltonian functions on two dimentional tori %, as Arnold shows in \cite{Ar:top}, 
 give rise %under generic assumptions
 to area preserving flows %on $\mathbb{T}^2$
 which can be decomposed into a finite number of components filled with periodic trajectories 
%, whose boundaries are given by separatrices of fixed points, 
and one ergodic component (see \cite{Ar:top}). %He also proves that t
The flow on this ergodic component is isomorphic to a suspension flow built over a rotation of the circle with the help of a roof function which has asymmetric logaritmic singularities (see also Section \ref{defns} for precise definitions). 

The question about mixing of such flows, risen in the same paper \cite{Ar:top}, was answered by Sinai and Khanin in \cite{SK:mix}, where it was proved that, under a generic diophantine condition on the rotation angle, suspension flows with asymmetric singularities over a rotation are strongly mixing (see also \cite{Ka:mix}). 
%Mixing in these flows is produced by different deceleration rates of neighbouring points, which have different return times. The phenomenon is sometimes referred as stretching of the Birkhoff sums (the geometric mechanism for mixing is intuitively explained in the following section). Mixing does not arise if the roof function has bounded variation, \cite{Ko:} since as proved by Kochergin in \cite{Ko:non} if the logarithmic singularities are symmetric z 
The diophantine condition of \cite{SK:mix} was weakened by Kochergin %in \cite{Ko:nonI} 
in a series of works (\cite{Ko:nonI, Ko:nonII, Ko:som, Ko:wel}).
%. In a further sequence of works (\cite{Ko:nonII, Ko:wel, Ko:som}) he also considers any irrational angle, provided a stronger assumption on the asymmetry of the roof funtion, and allows the presence of some symmetric singularities.
%, Kochergin relaxes the hypothesis further to cover any irrational angle, provided a stronger assumption on the asymmetry of the roof funtion, and allows the presence of some symmetric singularities.

Mixing in these flows is produced by different deceleration rates near the singular points. Neighbouring points on a Poincar\'e transversal have different return times and this causes a phenomenon sometimes called stretching of the Birkhoff sums (the idea of how this stretching leads to mixing is explained in Section \ref{geommech}). A similar stretching of Birkhoff sums was also used by Fayad in \cite{Fa:ana} to construct mixing reparametrization of flows on $\mathbb{T}^3$.  

Mixing does not arise in suspension flows over rotations %if there are no fixed points or 
in the case of bounded variation roof functions \cite{Ko:abs}. The presence of a symmetric logarithmic singularity is also not enough, as it was shown by Kochergin in \cite{Ko:non}. % if the logarithmic singularities are symmetric. 
Lema{\'{n}}czyk \cite{Le:sur} proved the absence of mixing if the Fourier coefficients of the roof function are of order $O(1/|n|)$ and showed with Fr{\c a}czek that these flows are disjoint in the sense of Furstenberg from all mixing flows \cite{FL:acl}. This condition is essentially sharp, see \cite{Ko:ami}. 
%A similar phenomenon of stretching of Birkhoff sums was also used by Fayad in \cite{Fa:} to construct mixing reparametrization of flows on $\mathbb{T}^3$. 

Consider, instead of $\mathbb{T}^2$, a compact orientable surface $M_g$ of higher genus ($g\geq 2$). A closed Morse one form $\omega$  
%and consider the same type of 
generates a Hamiltonian flow determined by the multivalued Hamiltonian $H$ locally defined by $\ud H = \omega$. %. As rotations arise as 
%As proven for example in \cite{Zo:how}, t
The corresponding area preserving flow on $M_g$ can be decomposed into %a finite number of 
components filled by periodic orbits and %a finite number of 
components on which the flow is metrically isomorphic to a suspension flow over an interval exchange transformation (IET) (see e.g. \cite{Zo:how}). Interval exchange transformations are piecewise orientation-preserving isometries of an interval which appear naturally as first return maps of such flows on a transversal, as rotations do in the case of $\mathbb{T}^2$. 

It was proved by Katok in \cite{Ka:int} that suspension flows over IETs under roof functions of bounded variation are never mixing and (see   \cite{FL:ond}) are disjoint from mixing flows. % (see %the paper by Fraczek and Lemanczyk \cite{FL:ond}).
On the other hand, Kochergin (see \cite{Ko:mix}) proves mixing for a class of roof functions over IETs which includes power-like singularities, which arise when the fixed points on the corresponding surface flow are degenerate. The presence of non-degenerate fixed points give rise to logarithmic singularities.
Fr{\c a}czek and Lema{\'{n}}czyk prove in \cite{FL:ond} that in the case of symmetric logarithmic singularities and typical IETs of $2$ or $3$ intervals % (which can be reduced to the study of rotations)
 the suspension flows are also disjoint from mixing flows. %Suspension flows with symmetric logarithmic singularities over rotations are also used by Fraczek and Lemanczyk \cite{FL:} to construct flows on some higher genus surface which are weakly mixing but not mixing.

In this paper we consider suspension flows over IETs of an arbitrary number of intervals with roof functions having a single asymmetric logarithmic singularity. We prove that for typical IETs such flows are strongly mixing. The case of several asymmetric singularities will be treated in another paper. 
%A result similar to ours was announced by Khanin and Nogueira in \cite{Zo:how}, but has never been actually published. 

As it was mentioned above the main mechanism of mixing is the stretching of Birkhoff sums. The proof of stretching in our case uses as a tool the Rauzy-Veech renormalization algorithm for IETs (see Section \ref{algorithmssec}). The condition on the IET which guarantees mixing is typical in view of a recent result in \cite{AGY:exp}. 
%and the proof of stretching for Birkhoff sums are based on the Rauzy-Veech renormalization algorithm for IETs (see Section \ref{algorithmssec}). %Conditions of diophantine type for IETs can be expressed in terms of frequencies of occurrences of positive matrices in the associated Rauzy-Veech cocycle. Various generic conditions with these flavour appear in the works by \cite{} 

\subsection{Definitions and Main Result.}\label{defns}
%We give now the definition of a suspension flow over an IET built under a given roof function.
\paragraph{Interval exchange transformations.}
Let $I^{(0)}=[0,1)$ and let $T:I^{(0)}\rightarrow I^{(0)}$ be an \emph{interval exchange transformation} (IET) of $d$ subintervals, i.e. a piecewise orientation preserving isometry of $I^{(0)}$ defined in the following way. Assign a permutation $\pi\in S_{d}$ and a partition of $I^{(0)}$ into $d$ subintervals, $I^{(0)}_1$, $I^{(0)}_2$, $\dots$, $I^{(0)}_{d}$, defined by a lengths vector $\underline{\lambda}=(\lambda_1, \lambda_2, \dots, 
\lambda_{d})$, $\lambda_i >0$, $\sum_{i=1}^{d}\lambda_i=1$, such that $\lambda_i=|I^{(0)}_i|$. Then $T$ permutes the subintervals according to $\pi$ so that under the action the transformation $I^{(0)}_i$ becomes the $\pi(i)$$^{th}$ interval, i.e. the order of the subintervals after applying $T$ is $I^{(0)}_{\pi^{-1}(1)}$,$I^{(0)}_{\pi^{-1}(2)}$ $\dots $ $I^{(0)}_{\pi^{-1}(d)}$. More precisely 
\begin{eqnarray}
I^{(0)}_{j} & \doteqdot &[ \sum_{i=1}^{j-1} \lambda_i , \sum_{i=1}^{j} \lambda_i [ \qquad j=1, \dots , d; \nonumber \\
T(x) & = & x  - \sum_{i=1}^{j-1} \lambda_i + \sum_{i=1}^{j-1} \lambda_{\pi^{-1}i} \quad \mathrm{for} \quad x \in I^{(0)}_{j}, \qquad j=1, \dots , d \nonumber . 
\end{eqnarray}
We shall often use the notation $T=(\lambda, \pi)$. 

\paragraph{Suspension flows.}
Let $f\in L^1 (I^{(0)}, dx)$ be a strictly positive function $f\geq m_f >0$ and assume $\int_{I^{(0)}} f(x) dx =1$. 
Further assumptions on $f$ will be formulated in Section \ref{1asymmetriclog}.
% and \ref{general}.
%In this section assume that $f\in \mathscr{C}^2(I^{(0)})$ and that there exists a positive constant C such that
%\be \label{log}
%\lim_{x\rightarrow 0+} \frac{f(x)}{|\log x|}=C,
%\ee
%i.e. $f$ has a \emph{one-sided logarithmic singularity} at the origin.
The \emph{phase space $X_f$} of the suspension flow is defined as
\bes
X_f \doteqdot \{ (x,y) |\quad  x \in I^{(0)}, \, 0\leq y < f(x) \}
\ees
and can be depicted as the set of points below the graph of the roof function $f$.
Introduce the normalized measure $\mu$ which is the restriction to $X_f$ of the Lebesgue measure $\ud x\,\ud y$.

The \emph{suspension flow built over $T$ with the help of the roof function f} is a one-parameter group $\{ \varphi_t \}_{t\in \mathbb{R}}$ of $\mu$-measure preserving transformations of $X_f$ whose action is generated by the following two relations:%acting according to:
\begin{equation} \label{suspflow}
 \left\{  \begin{array}{lll}
\varphi_t(x,y) &=& (x, y+t), \qquad \mathrm{if} \, 0\leq y+t < f(x); \\
\varphi_{f(x)}(x,0) &=& ( Tx,0).\\
		     \end{array} \right.
\end{equation}
Under the action of the flow a point of $(x,y) \in X_f$  moves with unit velocity along the vertical line up to the point $(x,f(x))$, then jumps instantly to the point $\left( T(x),0 \right)$, according to the base transformation. Afterwards it continues its motion along the vertical line and so on (see e.g. \cite{CFS:erg}).

We will denote %%%%%%ACCORCIABILE%the $r^{th}$ non-renormalized \emph{Birkhoff sum} of $f$ along the trajectory of $x$ under $T$ 
by\footnote{The dependence on $T$ is omitted when there is no ambiguity.}
\bes
\BS{f,T}{r}( x) = \BS{f}{r}( x)  \doteqdot \sum_{i=0}^{r-1} f(T^i(x)), \qquad x\in I^{(0)}. %,
\ees
the $r^{th}$ non-renormalized \emph{Birkhoff sum} of $f$ along the trajectory of $x$ under $T$. 

Given $x\in I^{(0)}$ denote by $r(x,t)$ the integer uniquely defined by
\be \label{defr}
r(x,t)\doteqdot \max \{ r\in \mathbb{N} \, | \quad  \BS{f}{r}(x) \leq t \},
\ee
which describes the number of \emph{discrete iterations} of the IET which the point $(x,0)$ undergoes before time $t$.
%
According to this notation the flow $\varphi_t$ defined by (\ref{suspflow}) acts as
\be \label{flowdef}
\varphi_t(x,0) = \left( T^{r(x,t)}(x), t- \BS{f}{r(x,t)}(x)\right).
\ee
For $t<0$, the action of the flow is defined as the inverse map. 

%\paragraph{Mixing.}
%Recall that the flow $\{ \varphi_t\}_{t\in \mathbb{R}}$ preserving the measure $\mu$ is said to be \emph{mixing} (or strongly mixing) if for each pair of measurable sets $A$, $B$, one has
%\be \label{mixingdef}
%\lim_{t\rightarrow \infty} \mu(\varphi_t(A)\cap B)=\mu(A)\mu(B).
%\ee
%Call \emph{rectangle} of base $I$ and height $h\leq m_f$ the set $R \doteqdot \Cup_{0\leq t\leq h}\varphi_{t}(I)$. Rectangles and their shifts $\varphi_t(R)$ generate the borel sigma-algebra of $(X_f,\mu)$.


\paragraph{Single asymmetric logarithmic singularity.} \label{1asymmetriclog}
Assume that $f\in \mathscr{C}^2\left((0,1)\right)$ and there exist two positive constants $C^+>0$, $C^- >0$, 
%$C^+ \neq C^-$, 
%%%%%%%%%%%%%%ELIMINABILE:sostituisci formato
such that
\be  \label{logsingforf''}
\lim_{x\rightarrow 0^+} \frac{f''(x)}{\frac{1}{x^2}}=C^+; \quad \qquad
 \lim_{x\rightarrow 1^-} \frac{f''(x)}{\left(\frac{1}{1-x}\right)^2}=C^- .\ee
%\begin{eqnarray} \label{logsingforf''}
%&& \lim_{x\rightarrow 0^+} \frac{f''(x)}{\frac{1}{x^2}}=C^+, \nonumber \\
%&&  \lim_{x\rightarrow 1^-} \frac{f''(x)}{\left(\frac{1}{1-x}\right)^2}=C^- \nonumber .
%\begin{eqnarray} \label{log}
%&& \lim_{x\rightarrow 0^+} \frac{f'(x)}{-\frac{1}{x}}=C+, \nonumber \\
%&&  \lim_{x\rightarrow 1^-} \frac{f'(x)}{\frac{1}{1-x}}=C^- \nonumber .
%%%\end{eqnarray}
It is easy to see that it implies that
\be\label{log}
\lim_{x\rightarrow 0^+} \frac{f(x)}{|\log x|}=C^+; \qquad \quad  \lim_{x\rightarrow 1^-} \frac{f(x)}{|\log (1-x) |}=C^- .
\ee
%\begin{eqnarray} \label{log}
%&& \lim_{x\rightarrow 0^+} \frac{f(x)}{|\log x|}=C^+, \nonumber \\
%&&  \lim_{x\rightarrow 1^-} \frac{f(x)}{|\log (1-x) |}=C^- \nonumber .
%\end{eqnarray}
Hence we say in this case that $f$ has a \emph{logarithmic singularity} at the origin. The singularity is called \emph{asymmetric} if $C^+ \neq C^-$. 

\paragraph{Mixing.}
Recall that a flow $\{ \varphi_t\}_{t\in \mathbb{R}}$ preserving the measure $\mu$ is said to be \emph{mixing} if for each pair of measurable sets $A$, $B$, one has
\be \label{mixingdef}
\lim_{t\rightarrow \infty} \mu(\varphi_t(A)\cap B)=\mu(A)\mu(B).
\ee

\paragraph{Main Result.}
The main result of this paper is the following.
\begin{thm}\label{mixing}
The suspension flow $\{\varphi_t \}_{t\in \mathbb{R}}$ built over a typical IET $T$ with the help of a roof function $f$ having a single asymmetric logarithmic singularity at the origin is mixing. \end{thm}
The notion of \emph{typical} IET is undestood from the measure theoretical point of view. More precisely, for every irreducible $\pi$, Theorem \ref{mixing} holds for a.e. lengths vector $\underline{\lambda}\in \Delta_{d-1}$ with respect to the Lebesgue measure on the simplex $\Delta_{d-1}$. 
%\begin{rem}
%A more precise characterizat is given in Section \ref{existencebalancedtimessec}.
%h\end{rem}

\subsection{A criterium for mixing.}\label{mixingcriteriumsec}
\paragraph{Partial partitions and rectangles.} By a \emph{partial partition} $\eta$ of $I^{(0)}$ into intervals we mean a collection of disjoint intervals $I=[a,b[$. We do not require that the union of these intervals is the whole $I^{(0)}$. All the partitions in this paper will be partial partitions into a finite number of intervals.
%A 1-parameter family of partial partitions $\eta(t)$ converges to $\eta$, trivial partition into points, and we write $\eta(t) \rightarrow \eta$ if for any $\delta >0$ and any set $C \subset I^{(0)}$ there is a $t_0$ such that for each $t\geq t_0$ there is a set $C_t$ consisting of elements of $\eta(t)$ (i.e. $\eta(t)$-measurable) such that $\mu(C \Delta C_t) < \delta$, where $\Delta$ denotes the symmetric difference.  
Denote by $Leb$ the Lebesgue measure on the Borel subsets of $I^{(0)}$. By using the notation $Leb(\eta)$ we mean the total measure of a partition  $\eta$, i.e.   
%%%\bes
$Leb(\eta) \doteqdot \sum_{I\in \eta} Leb(I)$.
%\ees
The \emph{mesh} of the partition $\eta$ is given by 
%%%\bes
$mesh(\eta) \doteqdot \sup_{I\in \eta }Leb(I) $.
%\ees
We will consider 1-parameter families of partial partitions $\eta(t)$, $t\in \mathbb{R}$.
%%%%%%%%%
%Consider a 1-parameter family of partial partitions $\eta(t)$. For any $\delta >0$, if $Leb(\eta(t)) > 1-\delta$ for each $t$ and $mesh(\eta(t))$ tends to zero as $t\rightarrow + \infty$, for any measurable set $C \subset I^{(0)}$ there exists $t_0$ such that for each $t\geq t_0$, there is a set $C_t$ consisting of a finite number of elements of $\eta(t)$ (i.e. $\eta(t)$-measurable) such that $\mu(C \Delta C_t) < 2\delta$, where $\Delta$ denotes the symmetric difference.

Call \emph{rectangle} of base $b(R)\subset I^{(0)}$ and height $h=h(R) < m_f$ the set $R$ of points $(x,y)$ such that $0\leq y\leq h$ and $x\in b(R)$. % \doteqdot \bigcup_{0\leq t < h}\varphi_{t}(b(R))$.  
 Rectangles and their shifts $\varphi_t(R)$ generate the Borel $\sigma$-algebra of  $(X_f,\mu)$.


%\subsection{A criterium for mixing}\label{mixingcriterium}
\paragraph{Mixing criterium.}
In order to show mixing it enough to verify the following criterium, similar to the one also used in \cite{Ko:mix, Fa:ana}.
\begin{lemma}[Mixing criterium] \label{mixingcriterium}
If, given any rectangle $R$, any $\epsilon>0$ and any $\delta >0$, one can find $t_0>0$ such that for each $t\geq t_0$ one can define a partial partition $\eta(t)$ of $I^{(0)}$ into intervals such that
%and there is a $t_0$ such that for each $t \geq t_0$ we can define a partial partition $\eta(t)$ of $I^{(0)}$ into intervals, in such a way that  
\be \label{partitiontendstopoint}
Leb(\eta(t))> 1-\delta, \qquad mesh (\eta(t) ) \leq \delta 
\ee
%and there exists a $t_0$ such that for each $t \geq t_0$ 
and for each $I\in \eta(t)$
%%%%%%%%%%%%%%%Original version check new one
%If for any rectangle $R$ and any $\epsilon>0$ there is a $t_0$ such that for each $t \geq t_0$ we can define a partial partition $\eta(t)$ of $I^{(0)}$ into intervals, in such a way that $\eta(t) \rightarrow \eta$ as $t \rightarrow \infty$ and for each $I\in \eta(t)$, , 
\be \label{mainestimate}
Leb (I \cap \varphi_{-t} (R)) \geq (1-\epsilon) Leb (I) \mu(R),
\ee
then the flow $\{ \varphi_t \}_{t\in \mathbb{R}}$ is mixing.
\end{lemma}
\begin{proof}
Mixing means that for any two measurable sets $A$ and $B$ and any $\epsilon>0$, for big enough positive $t$,
\be \label{mixingineq}
\mu(A\cap \varphi_{-t}(B))>(1-\epsilon)\mu(A)\mu(B),
\ee
since applying (\ref{mixingineq}) to $A^{C}$ and $B$ one gets 
\bes 
\mu(A \cap \varphi_{-t}(B))<(1+\epsilon)\mu(B)\mu(A)+ \epsilon\mu(B)
\ees
and therefore (\ref{mixingdef}). For $t<0$, it is enough to exchange the roles of $A$ and $B$ and use $\mu$-invarance of $\varphi_t$.
Moreover, it is enough to verify (\ref{mixingineq}) for $A$ and $B$ rectangles, since any measurable set can be approximated by a finite union of rectangles and their shifts under the flow.

Let $b(A)$ be the base of a rectangle $A$. For each $\delta>0$ and $t\geq t_0$,  
%by (\ref{partitiontendstopoint}) there exists $t_1>t_0$ is such that $mesh(\eta(t) )< \delta$ for $t\geq t_1$. For each $t\geq t_1$, 
there exists a finite number of intervals $I^{(t)}_{k} \in \eta(t)$, $k=0,\dots, K(t)$, such that
%, denoting by $\Delta$ the symmetric difference of sets, 
$Leb \left( b(A)\Delta \cup_{k=0}^{K(t)} I^{(t)}_k \right) \leq 3 \delta$. Here $\Delta$ denotes the symmetric difference of sets. To see this, consider all intervals of $\eta(t)$ which intersect $b(A)$ and use (\ref{partitiontendstopoint}).
Let 
\bes
A'\doteqdot  \bigcup_{0\leq y \leq h(A)} \left( \bigcup_{k_y=0}^{K(t+y)} I^{(t+y)}_{k_y}  \times \{ y \} \right).
\ees
Choosing $\delta \leq \frac{ \epsilon }{3} Leb (b(A)) \mu(B)$, %if $h(A)$ is the height of $A$, 
by Fubini theorem, 
%Choosing $\delta \leq \frac{ \epsilon }{3} Leb (b(A)) $, if $h(A)$ is the height of $A$, by Fubini theorem, 
%%%\bes
$\mu \left( A \, \Delta \, A' \right) \leq  \epsilon \mu(A)\mu(B)$ .
%\mu \left( A \, \Delta \bigcup_{0\leq y \leq h(A)} \left( \bigcup_{k_y=0}^{K(t+y)} I^{(t+y)}_{k_y}  \times \{ y \} \right) \right) \leq  \epsilon \mu(A) .
%\ees
%By (\ref{partitiontendstopoint}), choosing $\delta=  \epsilon/2 Leb \left( b_A\right) \mu(B)$,
%for each $\delta>0$, 
%when $t$ is big enough, there exists intervals $\{I_k |\, I_k\in \eta(t), k\in K\subset \mathbb{N} \}$ such that, if $b_A$ is the base of the rectangle $A$ and $h_A$ the height, 
%\bes
%Leb \left(  b_A \Delta \bigcup_{k\in K} I_k \right)< \epsilon Leb \left( b_A\right) \mu(B). 
%Hence, using Fubini again and 
Remarking the inequality $y\leq h(A) <  m_f$,  
% by definition of rectangle, 
we have, for each slice of $A'$,
\bes
\left( \cup_{k_y=0}^{K(t+y)} I^{(t+y)}_{k_y}   \times \{ y \} \right) \cap \varphi_{-t} (B) = \varphi_{y} \left( \left( \cup_{k_y=0}^{K(t+y)} I^{(t+y)}_{k_y}  \times \{0 \} \right) \cap \varphi_{-t-y} (B) \right) .
%\bigcup_{k_y=0}^{K(t+y)} I^{(t+y)}_{k_y}   \times \{ y \} = \varphi_{y} \left( \bigcup_{k_y=0}^{K(t+y)} I^{(t+y)}_{k_y}  \times \{0 \} \right)
\ees
Moreover $\varphi_y$ preserves $Leb$ on each slice and therefore one can assume that the hypothesis (\ref{mainestimate}) in which we set $B=R$ holds for all slices. Thus, combining these estimates and applying again Fubini theorem, we get, for $t\geq t_0$,
%\be
%\mu \left( \left( \bigcup_{0\leq y \leq h(A)} \bigcup_{k_y=0}^{K(t+y)} I^{(t+y)}_{k_y} \times \{ y \}   \right) \cap \varphi^{-t} (B) \right) \geq \int_{0}^{h(A)} (1-\epsilon)  Leb \left(  \bigcup_{k_y=0}^{K(t+y)} I^{(t+y)}_{k_y} \right) \mu(B) dy .
%\ee
\bes
\begin{split}
&\mu ( A \cap \varphi^{-t} (B))  \geq  
\mu \left( 
%\left( \bigcup_{0\leq y \leq h(A)} \bigcup_{k_y=0}^{K(t+y)} I^{(t+y)}_{k_y} \times \{ y \}   \right) 
A' \cap \varphi^{-t} (B) \right) - 3\delta h(A)
%\mu \left( ( h(A) \times \bigcup_{k_y=0}^{K(t+y)} I^{(t+y)}_{k_y} ) \cap \varphi^{-t} (B)\right)
\geq  \\ & \int_{0}^{h(A)} (1-\epsilon)  Leb %\left(
  \bigcup_{k_y=0}^{K(t+y)} I^{(t+y)}_{k_y} %\right)
\mu(B) dy - \epsilon \mu(A) \mu(B) 
%h(A) \int_{0}^{h(A)}  Leb \left(  \bigcup_{k_y=0}^{K(t+y)} I^{(t+y)}_{k_y} \right) 
%Leb(\bigcup_{k\in K} I_k ) 
 \geq \left ((1-\epsilon)^2 - \epsilon 
\right) \mu(A) \mu(B),
\end{split}
\ees
%%%%ELIMINABILE
hence proving the lemma.
\end{proof}

%\subsection{Construction of the mixing partition.}
\paragraph{Intuitive explanation of the mixing mechanism.} \label{geommech}
%%%ELIMINABILE
In this section we explain the geometric mechanism that produces mixing for this type of suspension flows. %The same mechanism, sometimes referred to as \emph{stretching of Birkhoff sums}, was used in \cite{SK:mix, Ko:nonI, Ko:nonII, Fa:ana}.
\begin{figure}
\centering
\subfigure[ $\varphi_t(I)$ ]{\label{verticalstripg}
\includegraphics[width=0.55\textwidth]{logflows.eps}}
\hspace{5mm}
\subfigure[ Toy model ]{\label{toyflow}
\includegraphics[width=0.3\textwidth]{toyflow.eps}}
%\caption{The image $\varphi_t(I)$ of a small segment $I$.}
%\label{verticalstripg}
\end{figure}
%%%%RIMPICCIOLISCI FIGURA E TOGLI SPAZIO qui dopo

Consider a sufficiently small segment $I=[a,b]\subset I^{(0)}$ and let us understand how its image under the flow, $\varphi_t(I)$, looks like for very large $t$. We claim that $\varphi_t(I)$ will consist of many almost vertical curves, as shown in Figure \ref{verticalstripg}. 

Assume as a toy example that $f$ has only a one-sided logarithmic singularity at the origin and is monotonically decreasing, as in Figure \ref{toyflow}. Notice first that until $t < m_f$, $\varphi_t(I)$ is still a horizontal segment, while as $t= f(x_0) $ for some $x_0\in I$, $\varphi_t(I)$ splits into two curves: one is a still an horizontal segment, while the other piece will project over $T ([x_0,b])$ and be a translate of the graph of $-f|_{[x_0,b]}$, as can be seen by (\ref{flowdef}). See Figure \ref{toyflow}. More generally, from (\ref{flowdef}), each of the curves in which $\varphi_t(I)$ will split is a graph of a translate of the Birkhoff sum $\BS{f}{r}$ restricted over a small interval of the form $T^{r}([x_i,x_{i+1}))$, where $[x_i,x_{i+1}) \subset I$. Noticing that $f'<0$ and the integral of $f'$ is divergent, one can prove in this toy model that the slopes of these curves, which are given by $-\BS{f'}{r}$, are growing to infinity, i.e. they are becoming almost vertical. Hence the increasingly big delay between different points causes $\varphi_t(I)$ to split into many curves, which are distributed over the orbit of $T^n(I)$. Using unique ergodicity of $T$ on the base and the fact that each strip can be approximated by a straight line, one can show that the fraction of $x\in I$ such that $\varphi_t(x)\cap R \neq \emptyset$ is proportional to $\mu (R)$.

When the singularity is asymmetric the same phenomenon happens and one can show that delays accumulated from visits on one side are stronger than opposite delays accumulated from the other, causing $\BS{f'}{r}(x)$ to diverge as in the presence of a one-sided singularity 
%with constant $(C^+-C^-)$
 for most of the points. 

\paragraph{Outline of the proof of Theorem \ref{mixing}.}
In order to prove mixing for the suspension flow, we use the criterium in Lemma \ref{mixingcriterium}. Given a rectangle $R$ and $\epsilon , \delta >0$,
our goal is to construct, for any sufficiently large $t$, a partial partition $\eta(t)$ of $I^{(0)}$ into intervals which satisfy (\ref{partitiontendstopoint}, \ref{mainestimate}). Each of the intervals of these partitions behaves under the flow as explained in the previous paragraph. The construction of the partition is carried out in several steps, explained in Section \ref{mixingpartitionsec}.
%s \ref{continuitysec}, \ref{restimatessec}. 
In order to get the final estimate (\ref{mainestimate}), in Section \ref{areaestimatesec}, the key step is to get a good estimate of the rate of growth of the first two derivatives of $\BS{f}{r}$. Such estimates, presented in Section \ref{growthBSsec}, are based on some property of the renormalization cocycle for IETs introduced by Rauzy, Veech and Zorich. The definition and some properties of this cocycle are recalled in Section \ref{algorithmssec}.


%%%%%%%%%

\section{Renormalization algorithms for IETs.} \label{algorithmssec}
Rauzy, Veech and Zorich (see \cite{Ra:ech, Ve:pro,{Zo:fin}}) developed a renormalization algorithm for IET which is a multi-dimensional generalization of the continued fraction algorithm. 
%This algorithm is the main tool in proving the estimates on the growth of the Birkhoff sums.
%Let us recall that this cocycle was also used in recent works by Avila, Bufetov, Forni, Gouezel, Marmi, Moussa and Yoccoz, see \cite{}.
%Recently Marmi, Moussa and Yoccoz \cite{}, Bufetov \cite{}, Avila and Forni \cite{} and Avila, Gouezel and Yoccoz \cite{} proved refined estimates for accelerated versions of the cocycle and used   This algorithm is the main tool in proving the estimates on the growth of the Birkhoff sums.

In what follows let $T=(\underline{\lambda}, \pi)$ be an IET. We assume that $\pi$ is \emph{irreducible}, i.e. if the subset $\{ 1, 2, \dots i \}$ is $\pi$-invariant, then $i=d$, since this is a necessary condition for minimality.
We also assume that $T$ satisfies the Infinite Distinct Orbit Condition (IDOC) introduced by M. Keane in \cite{Ke:int}. 
We say that $T$ has the (IDOC) if, denoting by $\beta_0 =0$ and $\beta_j \doteqdot \sum_{i=1}^{j} \lambda_i$ for $j=1, \dots , d$
 the discontinuities of $T$, the orbits $\mathscr{O}(\beta_j) \doteqdot \{ T^{n}(\beta_j) |\, n \in \mathbb{N}\}$, $1\leq j\leq d-1$, are infinite and disjoint, i.e. $\mathscr{O}(\beta_j)\cap \mathscr{O}(\beta_i)=\emptyset$ for any $i\neq j$. 
%$1\leq j\neq i\leq d-1$. 
As it was shown by Keane in \cite{Ke:int}, the (IDOC) implies minimality.

\subsection{Rauzy-Veech and Zorich algorithms and cocycles.}\label{induction}
Starting with $T=T^{(0)}$, the Rauzy-Veech algorithm produces a sequence of IET $T^{(r)}$ which are induced maps of $T$ onto a sequence of nested subintervals $I^{(r)}\subset I$. It is easy to see that in general the induced first return map of $T$ on a subinterval $I'\subset I$ is again an IET, of at most $d + 2$ intervals. One Rauzy-step is defined so that $T^{(1)}$ is an exchange of exactly the same number $d$ of subintervals.

%%%ELIMINABILI Titoletti per accorciare...
\subsubsection{The induction algorithms.}
\paragraph{One step of Rauzy-Veech algorithm.}
At the first step, compare the lengths of $I_{d}$ and $I_{\pi^{-1}(d)}$, i.e. of the last subintervals before and after the transformation. It follows from the (IDOC) that $\lambda_{d} \neq \lambda_{\pi^{-1}d}$. Hence there can be two cases:
\begin{itemize}
\item[(a)]
$\lambda_{d} < \lambda_{\pi^{-1}d}$. In this case we consider the new interval $I^{(1)}\doteqdot [0, 1-\lambda_{d}[$. Define $T^{(1)}$ to be the induced map, i.e. the first return map of $T^{(0)}$ onto $I^{(1)}$. It is important that it is again an IET of the same number $d$ of exchanged intervals.

\item[(b)]
$\lambda_{d} > \lambda_{\pi^{-1}d}$. In this case we consider the new interval $I^{(1)}\doteqdot [0, 1-\lambda_{\pi^{-1}d}[$ and, as before, define $T^{(1)}$ to be the induced map on $I^{(1)}$. Also in this case $T^{(1)}$ is again an IET of $d$ intervals.
\end{itemize}

Since $T^{(1)}$ is again an exchange of $d$ intervals, we can write $T^{(1)}=(\lambda^{(1)}, \pi^{(1)})$ (see Figure \ref{figTa},\ref{figTb}), defining in this way a new lengths vector and a new permutation.
%%%%%ELIMINABILE con riferimenti
\begin{figure}
\centering
\subfigure[Case $\lambda_{d} < \lambda_{\pi ^{-1}(d)}$]{\label{figTa}\includegraphics[width=0.44\textwidth]{figTa.eps}}
\hspace{5mm}
\subfigure[Case $\lambda_{d} > \lambda_{\pi ^{-1}({d})}$]{\label{figTb}\includegraphics[width=0.44\textwidth]{figTb.eps}}
\caption{One step of Rauzy-Veech algorithm on $T=(\lambda, (4321))$.}
\label{passoVeechfig2}
\end{figure}
One can explicitly write the expressions for two combinatorial operators $a$ and $b$ on $\Sym{d}$, where $\Sym{d}$ is the space of permutations of $d$ elements, such that $\pi^{(1)} = a \pi$ or $ b \pi$ respectively. Explicitly:
%%%%ELIMINABILE Mettere due in uno
\bes
%\bes \begin{array}{ll}
a\pi (j)=  \left\{  \begin{array}{ll}
			\pi (j) & j\leq \pi^{-1}(d) ; \\
			\pi (d) & j = \pi^{-1}(d) + 1 ;\\
			\pi (j-1) & \mathrm{otherwise} ;  \\
		     \end{array} \right.
\ees
\bes
%&
b\pi (j)=  \left\{  \begin{array}{ll}
			\pi (j) & j\leq \pi(d) ; \\
			\pi (j) + 1 & \pi(d) < \pi (j) < d ;\\
			\pi (d) + 1 & \pi (j) = d . \\
		     \end{array} \right.
%\end{array} \ees
\ees
%%%ELIMINABILE Accorciare
Introduce the following matrices to describe the new lengths. Denote by {\it Id}
 the identity $d \times d$ matrix and by $E_{i,j}$ the matrix whose only non zero entry is $(E_{i,j})_{ij}=1$. Introduce the auxiliary permutation $\tau_s \in S^{d}$, $\tau_s = (1\, \, 2\, \, \dots \, s \, \,  s\!+\!2\, \, \,  \dots \, \,  d\, \, s \!+\!1 )$ if $ 1\leq s <d-1$ and $\tau_{d-1} =   \mathrm{id}$, which rotates cyclically all elements after the $s^{th}$-one. Denote by $P(\tau_s )$ the matrix associated to the permutation, i.e. $P(\tau_s )_{ij}=\delta _{i \tau_s(i)}$.
The two \emph{Rauzy-Veech elementary matrices} associated to $(\underline{\lambda},\pi)$ are defined by
\be\label{matriciaeb}
\left\{  \begin{array}{rcl}
			A(\pi ,a) &= & ({\it Id} + E_{\pi ^{-1}(d),d} ) \cdot P(\tau _{\pi ^{-1}(d)} ) ; \\
			A(\pi ,b) &= & {\it Id} + E_{d,\pi^{-1}(d)} . \\
		     \end{array} \right.
\ee
%The matrix $A(\pi ,b)$ has only one $1$ out of diagonal, at entry $(\pi ^{-1}(d),d)$, while $A(\pi ,a)$ is obtained by an analogous matrix with only one non zero entry at $(d,\pi^{-1}(d))$ by shifting cyclically all columns after the $\pi^{-1}(d)^{th}$-one. Notice that both matrices have non negative entries and determinant one. 
%\[ \begin{array}{ccc}
% A\left(\pi,b\right) & A \left( \pi,a \right) \\
%\left( \begin{array}{cccccc}
%			1&&&&&0 \\
%			&1&&&& \\
	%		&&\ddots&& \\
		%	&&&\ddots& \\
			%&&&&1& \\
			%0&&1&&&1 \\
%	\end{array} \right) & \left( \begin{array}{cccccc}
	%			1&&&&&0 \\
		%		&\ddots&&&& \\
			%	&&1&1&& \\
				%&& & 0&\ddots& \\
				%&&&&\ddots&1 \\
				%0&&1&&&0 \\
		    		 %\end{array} \right) 
                           % & 
%\begin{array}{cc}
	%			& \\
		%		& \\
			%\leftarrow	& \pi^{-1}(d) \\
				%& \\
				%& \\
				%& \\
		    		% \end{array} 

%\\
% \begin{array}{cccccc}	  && \uparrow &&&  		 \end{array} & \begin{array}{cccccc}	  && \uparrow &&&  		 \end{array} &  \begin{array}{cc}	  &  		 \end{array} \\
%  \begin{array}{cccccc}	  && \pi^{-1}(d)&&&  		 \end{array} & \begin{array}{cccccc}	  && \pi^{-1}(d)&&&  		 \end{array} &  \begin{array}{cc}	  &  		 \end{array} \\
%\\
%\end{array} \]
\noindent The induced IET $T^{(1)}$ is then given by
\begin{equation} \label{T1}
(\underline{\lambda}^{(1)} , \pi^{(1)} ) \doteqdot \left\{  \begin{array}{ll}
			\left( A^{-1}(\pi , \, a)\cdot \underline{\lambda} ,\, a(\pi ) \right), & \lambda_{d} < \lambda_{\pi^{-1}({d})} ; \\
			\left( A^{-1}(\pi , \, b)\cdot \underline{\lambda} ,\, b(\pi ) \right), & \lambda_{d} > \lambda_{\pi^{-1}({d})} . \\	
		     \end{array} \right.
\end{equation}
%The matrix $A(\pi ,a)$ has a more complicated form then $A(\pi ,b)$ because subintervals following $I_{\pi^{-1}(d)}$ undergo a relabeling in the new IET, the $i^{th}$ becoming the $(i+1)^{th}$ (when $i \neq d$). See e.g. Fig. \ref{figTa}.
Remark that both $A(\pi ,a)$ and $A(\pi ,b)$ belong to $SL(d, \mathbb{Z})$ and have non-negative entries.

%%%Spazio
Define inductively $T^{(r)}=(\underline{\lambda}^{(r)}, \pi^{(r)})$ to be the induced map of $T^{(r-1)}$ on $I^{(r)}$. It can be seen that the (IDOC) assures that the algorithm is well defined at each step, i.e. that $\lambda^{(r-1)}_{d} \neq \lambda^{(r-1)}_{ (\pi^{(r-1)} ) ^{-1}(d)}$ for any $r \in \mathbb{N}$.


\paragraph{Renormalized Rauzy-Veech map.}
% and Zorich acceleration.}
The \emph{Rauzy class} of $\pi$, denoted by $\mathscr{R}(\pi)\subset S_{d}$, is the set of all permutations obtained iterating the operators $a$ and $b$ starting from $\pi$. % (not all permutations of $S_{d}$ can be obtained by iterating the algorithm).
Using the norm $|\underline{\lambda}|=\sum_{i=1}^{d}\lambda_i$, assume that the initial lengths belong to the simplex $\Delta_{d-1}$ of vectors $\underline{\lambda} \in \mathbb{R}^{d}_{+}$ such that $|\underline{\lambda}|=1$. 
Let us denote by $\Delta(\mathscr{R}) = \Delta_{d-1} \times \mathscr{R}(\pi)$ the space of IETs on the unit interval corresponding to a given Rauzy class $\mathscr{R}$. 

Consider the map on  $\Delta(\mathscr{R})$
%the space $\Delta_{d-1} \times \mathscr{R}(\pi)$ of IETs on the unit interval, 
which associates to $T$ the induced IET after one step of the algorithm including the following renormalization:
\bes \R \left ( (\underline{\lambda}^{(0)},\pi^{(0)}) \right) \doteqdot \left( \frac{\underline{\lambda}^{(1)}}{|\underline{\lambda}^{(1)}|},\pi^{(1)} \right).
\ees 
Let us call it the \emph{Rauzy-Veech map}. 
Veech proved that $\mathscr{R}$ admits an invariant measure $\mu_{\V}$, absolutely continuous with respect to the Lebesgue measure, which is infinite. 
The main result proved by Veech in \cite{Ve:gau} is that the map $\R$ is conservative. As a consequence, he proves that given $\pi \in \Sym{d}$, for a.e. $\underline{\lambda} \in \Delta_{d-1}$, the IET $T=(\underline{\lambda}, \pi)$ is uniquely ergodic.

%The main results proved by Veech in \cite{Ve:gau} are the following. Theorem \ref{Veechuniquelyergodic} is based on Theorem \ref{Veechergodic}. 
%\begin{thm}[Veech]\label{Veechergodic}
%The map $\R: $ is ergodic with respect to the invariant measure $\mu_{\V}$. 
%\end{thm}
%\begin{thm}[Veech]\label{Veechuniquelyergodic}
%Given $\pi \in \Sym{d}$, for a.e. $\underline{\lambda} \in \Delta_{d-1}$, the IET $T=(\underline{\lambda}, \pi)$ is uniquely ergodic.
%\end{thm}

%%%ELIMINABILE?? Is it ever needed? The measure yes...
\paragraph{Zorich acceleration.}
% and Zorich acceleration.}
Take an IET $T$ and consider its Rauzy-Veech orbit $\{\R^n T \}_{n\in \mathbb{N}}$. 
%$T, \R T, \R^2 T, \dots \,$. 
In a typical situation one can find an integer $z_o=z_o(T)>0$ so that $T, \R T,  \dots , \R^{z_0-1}(T) $ all correspond to the same case $(a)$ or $(b)$ while $\R^{z_o}(T) $ corresponds to the other one. Grouping together these $z_0$ steps of Rauzy induction, we get a new transformation $\Z$ on the space of IET, where the letter $\Z$ is chosen in honor of A. Zorich who introduced this map in \cite{Zo:fin}. %From the point of view of general ergodic theory $\Z$ is the result of changing velocity of $\R$. 
Zorich showed in \cite{Zo:fin} that $\Z$  has an absolutely continuous \emph{finite} invariant measure. 
We will denote the \emph{Zorich invariant measure} by $\mu_{\Z}$ . 
%When the permutation $\pi$ is clear, we will often use $\mu_{\Z}$ to denote the restriction of the measure to  $\Delta_{d-1} \times \{ \pi \}$, writing $\mu_{\Z}(\Delta)$ for some subset $\Delta \subset \Delta_{d-1}$ of initial lengths instead than $\mu_{\Z}(C \times \{ \pi \})$.

%\paragraph{Rauzy-Veech lenghts cocycle.}
\subsubsection{Rauzy-Veech lengths cocycle.}
As was explained above, to each $T$ one can associate an elementary matrix $A(T)$ in $SL(d, \mathbb{Z})$ defining $A(T)\doteqdot A(\pi ,a)$ or $A(T)\doteqdot A(\pi ,b)$ respectively. 
Let $\RL{r} = \RL{r} (T) \doteqdot  A(\R^r T)$. Then for each $r$ we can associate to $T$ the product:
\bes
\RLp{r} \doteqdot \RL{1} \dots \RL{r}. 
\ees
We can easily see that the map  $A^{-1}$: $\Delta(\mathscr{R}) \rightarrow SL(d,\mathbb{Z})$ is a cocycle over $\R$, which we call the \emph{Rauzy-Veech lengths cocycle}.
Iterating the lengths relation in (\ref{T1}) we get the formula for the lengths vector of $T^{(r)}$:
\begin{equation} \label{lengthsrelationR}
\underline{\lambda} ^{(r)} = \left( \RLp{r} \right) ^{-1 }  \underline{\lambda} .
\end{equation}

Let us also introduce the following notation useful to consider more generally products of Rauzy-Veech cocycle matrices from $m$ to $n$, $m<n$:
 \bes \RLp{m,n} \doteqdot \RL{m} \cdot  \RL{m+1} \cdot \, \dots \, \cdot \RL{n-2} \cdot \RL{n-1}.
\ees

%%%%%%One can define also the accelerated \emph{Zorich lengths cocycle} $\ZLs^{-1}$: $\Delta(\mathscr{R}) \rightarrow SL(d,\mathbb{Z})$ over the Zorich map $\Z$, given by $\ZLs (T) \doteqdot \RLp{z_o(T)}$. 
%As for the Rauzy-Veech cocycle, we set $\ZL{n} \doteqdot \ZL{Z^{n}T}$
%\be
%\ZLp{n} \doteqdot \ZL{1} \dots \ZL{n} =  \RLp{z_o(T)}(T)  \RLp{z_o(\Z T)}(\Z T)  \dots \RLp{z_o(\Z^{n-1} T)}(\Z^{n-1} T).
%\ee
%%%%%%One of the main advantages in using Zorich acceleration is that, as proved by Zorich in \cite{Zo:fin}, the accelerated cocycle $\ZLs$ over the space $\Delta(\mathscr{R})$ of IETs is \emph{integrable}, i.e.
%%%\bes
%%%\int_{\Delta} \log \|\ZLs{(T)} \| \ud \mu(T) < + \infty.
%%%\ees
%the following result proved in \cite{Zo:fin}.
%\begin{thm}[Zorich]\label{Zorich}
%The accelerated cocycle $\ZLs$ over the space of IETs $\Delta(\mathscr{R})$ is \emph{integrable}, i.e.
%\bes
%\int_{\Delta} \log \|\ZLs{(T)} \| \ud \mu(T) < + \infty.
%\ees
%\end{thm}

%If $n=n(T)=z_0(T)+ z_0(\Z T) + \dots z_0(\Z^{n-1} T)$, then $\underline{\lambda}^{(n(T))}$ is the lengths vector corresponding to $\Z^{n} T$ before renormalization. The lengths vectors relation (\ref{lenghtsrelationR}) becomes
%\be \label{lengthsrelationZ}
%\underline{\lambda} ^{(n)} = \left( \ZLp{n} \right) ^{-1 }  \underline{\lambda} .
%\ee

%\paragraph{Notation for Zorich time.}
%From now on, by abusing the notation, we will write simply $\underline{\lambda}^{(n)}$. When the index $n$ is used, we will always refer to Zorich time $n=n(T)$. Similarly $T^{(n)}$, with $n=n(T)$, will always denote the induced map obtained considering the $n^{th}$ step of the accelerated Zorich algorithm. 


%\paragraph{Occurrences of cocycle product matrices.}
%Let us also introduce the following notation useful to consider more generally products of Rauzy-Veech cocycle matrices from $m$ to $n$, $m<n$:
% \begin{equation} \ZLp{m,n} \doteqdot \ZL{m} \cdot  \ZL{m+1} \cdot \, \dots \, \cdot \ZL{n-2} \ZL{n-1}.
%\end{equation}

%The following result, which we need later, is contained in \cite{Ve:gau}.
%\begin{thm}[Veech]\label{Veech}
%For a.e. IET $T=(\lambda, \pi)$ 
%if $B=\ZLp{n,m}(T)$ for some $n,m \in \mathbb{N}$, the matrix  $B$ occurs infinitely many times as product of consecutive elements of the sequence $\{ \ZL{n} \}_{n\in \mathbb{N}}$.
%\end{thm}

%We need to remind the following definitions to state Proposition \ref{existencebalancedtimes}.
%\paragraph{Hilbert metric and projective contractions.} \label{Hilbertsec}
\subsubsection{Hilbert metric and projective contractions.} \label{Hilbertsec}
Consider on the simplex $\Delta_{d-1} \subset \mathbb{R}_+^d$ the \emph{Hilbert distance} $d_H$, defined as follows.
\be
d_H(\lambda, \lambda') \doteqdot \log \frac{\max_{i=1,\dots , d}\frac{\lambda_i}{\lambda'_i}}{\min_{i=1 , \dots ,d}\frac{\lambda_i}{\lambda'_i}}.
\ee
We denote the diameter with respect to $d_H$ of a projective subset $\Lambda \subset \Delta_{d-1}$ by
%%%ELIMINABILE formula
\be \label{diameter}
diam_H (\Lambda ) \doteqdot \sup_{\lambda, \lambda' \in \Lambda} d_H( \lambda, \lambda' ) .
\ee
Remark that if its closure $\overline{\Lambda} \subset \Delta_{d-1}$, 
%where $\overline{\phantom{\Lambda}}$ we denote the closure,
  then $diam_H (\Lambda )$ is finite. 

Let us write $A\geq 0$ if $A$ has non negative entries and $A>0$ is $A$ has strictly positive entries.
Recall that to each $A\in SL(d,\mathbb{Z})$, $A\geq 0$, one can associate a projective transformation $\widetilde{A}: \Delta_{d-1} \rightarrow \Delta_{d-1}$ given by
\bes
\widetilde{A} \lambda = \frac{A \lambda }{|A \lambda |}.
\ees
When $A \geq 0$, $d_H(\widetilde{A} \lambda, \widetilde{A} \lambda' ) \leq d_H(\lambda, \lambda')$. Furthermore, if $A>0$, then we get a contraction. More precisely, $A>0$ is equivalent to the closure $ \overline{ \widetilde{A} \left( \Delta_{d-1} \right)}$ being contained in $\Delta_{d-1}$, hence defining 
\be \label{D_Adef}
D(A) \doteqdot diam_H \left( \, \overline{ \widetilde{A}  \left( {\Delta}_{d-1} \right) }\, \right) , 
\ee
we have $D(A) < \infty$. Then
\be \label{contraction}
 d_H(\widetilde{A} \lambda, \widetilde{A} \lambda' ) \leq (1-e^{-D(A)}) d_H(\lambda, \lambda').
\ee

%as proven for example in \cite{Vi:lec}, let  
%\be \label{contraction}
%D_A \doteqdot \sup_{\lambda, \lambda' \in \Delta_{d-1}} d_H(\widetilde{A} \lambda,\widetilde{A} \lambda' ).
% < \infty ,
%\sup \left( \log \left( \frac{\max_{i=1\dots d}\frac{\sum_j A_{ij} \lambda_j}{\sum_j A_{ij} \lambda_j'}}{\min_{i=1\dots d}\frac{\sum_j A_{ij} \lambda_j}{\sum_j A_{ij} \lambda_j'}} \right) \right) < \infty ,

%Since $A>0$, $D_A$ is finite. 
%then $d_H(\widetilde{A} \lambda, \widetilde{A} \lambda' ) \leq (1-e^{-D}) d_H(\lambda, \lambda')$.

\paragraph{Paths on Rauzy classes.}\label{rauzyclasses}
Rauzy classes can be visualized in terms of directed labeled graphs, the \emph{Rauzy graphs}. Vertices are in one-to-one correspondence with permutations of  $\mathscr{R}(\pi)$; arrows connect permutations obtained one from the other by applying $a$ or $b$ and are labeled according to the type, $a$ or $b$ respectively. 
Each vertex is the starting point and the ending point of exactly two arrows, one of each type. We will denote by $\gamma_i=\gamma_i(\pi', a)$ ($\gamma_i(\pi', b)$) the arrow of type $a$ (type $b$) coming out from the vertex $\pi'$. 
 
A \emph{path} $\gamma = (\gamma_1, \dots ,\gamma_r)$ is a sequence of compatible arrows on the Rauzy graph, i.e. such that the starting vertex of $\gamma_{i+1}$ is the ending vertex of $\gamma_{i}$, $i=1,\dots, r-1$. 
%A path $\gamma$ is uniquely defined by its starting point $(\nu_0,\nu_1)$ and by the sequence of types $a$ and $b$ associated to $\gamma_{i}$, $i=1,\dots, n$, and will be also denoted by $\gamma=(\nu_0,\nu_1, t_1, \dots, t_n)$ where $t_i \in \{ a,b \}$ are the corresponding types. 
Given a path $\gamma$, we can associate to it a matrix 
\bes
A(\gamma) \doteqdot  A(\gamma_1)\cdot \, \dots \, \cdot A(\gamma_r),
\ees
where  $A(\gamma_i)= {A(\pi' , a)}$ if $\gamma_i=\gamma_i(\pi_i, a)$ and ${A(\pi_i, b)}$ if $\gamma_i=\gamma_i(\pi_i, b)$. Associate to $\gamma$ also the subsimplex:
\be
\Delta(\gamma) \doteqdot \left\{ \widetilde{A(\gamma)} \, \underline{\lambda}\, | \quad  \underline{\lambda} \in \Delta_{d-1}  \right\} \subset \Delta_{d-1} .
%\Delta(\gamma) \doteqdot \left\{ \frac{A(\gamma) \cdot    \underline{\lambda} }{|A(\gamma) \cdot  \underline{\lambda}|} :  \underline{\lambda} \in \Delta_{d-1}  \right\} .
\ee
Using the induction, one can easily verify the following.
\begin{rem}\label{sequence}
If $T=(\underline{\lambda}, \pi)$ and $\underline{\lambda} \in \Delta(\gamma)$ where $\gamma= (\gamma_1,\dots, \gamma_r)$ is a path starting at $\pi$, $\gamma_i =\gamma_i (\pi_i,c_i)$, $c_i\in\{ a,b \}$,  
%where $\pi$ is the starting vertex of $\gamma$ and $\underline{\lambda} \in \Delta(\gamma)$, %, one can easily prove by induction that 
the sequence of types and permutations obtained in the first $r$ steps of Rauzy-Veech induction is determined by $\gamma$, %the sequence of types and starting vertices $\pi_i$ of the arrows $\gamma_1,\dots ,\gamma_r$,
 i.e. $\RL{i}(T)=A(\gamma_i)$ and $\pi^{(i)} = \pi_i$. 
\end{rem}

%\paragraph{The natural extension.}
\subsubsection{The natural extension.}
The natural extension $\hat{\R}$ of the map $\R$ was introduced by Veech \cite{Ve:gau} and admits a geometric interpretation in terms of the space of zippered rectangles. %We refer to Veech \cite{Ve:gau} or  \cite{MMY:coh} for the construction of zippered rectangles. 
We use the simpler choice of coordinates for zippered rectangles, adopted by \cite{Bu:dec, MMY:coh, AGY:exp}. 

%%%
Consider the following polyhedral cones $\Theta_{\pi}\subset \mathbb{R}^{d}$, where $\pi \in \mathscr{R}$.% and $\underline{\tau}=(\tau_1,\dots ,\tau_d) \in \mathbb{R}^{d}$:
\bes
\Theta_{\pi} \doteqdot \{ \underline{\tau}=(\tau_1,\dots ,\tau_d)  \in   \mathbb{R}^{d}| \quad \sum_{i=1}^{k} \tau_i >0 , \, \sum_{i=1}^{k} \tau_{\pi^{-1}i} <0, \, k=1,\dots, d-1 \},
\ees
%Let $\Theta(\mathscr{R}) \doteqdot\bigcup_{\pi \in \mathscr{R}} \Theta_{\pi}$.
%Let $\Delta(\mathscr{R})\times \Theta(\mathscr{R}) \doteqdot \bigcup_{\pi \in \mathscr{R}} \Delta_{d-1} \times \{\pi\} \times \Theta_{\pi} $.
which is non-empty since if $\tau_i\doteqdot \pi(i) -i$, $\underline{\tau}\in \Theta_{\pi}$.

The real valued function $Area(\cdot)$ associates to $(\underline{\lambda}, \pi, \underline{\tau}) \in   \Delta_{d-1} \times \{\pi\} \times \Theta_{\pi}$, 
\bes
Area(\underline{\lambda}, \pi, \underline{\tau} ) \doteqdot \sum_{k=1}^{d} \lambda_k \, \left( \sum_{i=1}^{k-1} \tau_i  - \sum_{i=1}^{\pi(k)-1} \tau_{\pi^{-1}i}  \right). 
\ees
$Area(\cdot)$ has a geometric interpretation as the area of the zippered rectangle associated to the datas $(\underline{\lambda}, \pi, \underline{\tau})$ (see e.g. \cite{MMY:coh}). 
%one can define a real valued function 

Consider the following space as domain of the natural extension.
\bes
\hat{\Upsilon}^{(1)}_{\mathscr{R}}\doteqdot \{ (\underline{\lambda}, \pi, \underline{\tau} ) | \quad (\underline{\lambda}, \pi) \in \Delta(\mathscr{R}), \, \underline{\tau} \in \Theta_{\pi} ,\, Area((\underline{\lambda}, \pi, \underline{\tau} ) ) = 1   \}.
\ees
The map $\hat{\R}:\hat{\Upsilon}^{(1)}_{\mathscr{R}} \rightarrow  \hat{\Upsilon}^{(1)}_{\mathscr{R}} $ is defined as follows.% whenever $\R$ is defined.%
\footnote{More precisely $\hat{\R}$ is defined on triples $(\underline{\lambda}, \pi, \underline{\tau} )$ such that $(\underline{\lambda}, \pi)$ belong to the domain of $\R$.}
\bes
\hat{\R} \left( (\underline{\lambda}^{(0)}, \pi^{(0)}, \underline{\tau}^{(0)} ) \right) = \left( \R (\underline{\lambda}, \pi),  |\underline{\lambda}^{(1)}| \underline{\tau}^{(1)}  \right) = \left( \frac{\underline{\lambda}^{(1)}}{|\underline{\lambda}^{(1)}|},\pi^{(1)} ,|\underline{\lambda}^{(1)}|\underline{\tau}^{(1)}  \right) ,
\ees
where $(\underline{\lambda}^{(1)},\pi^{(1)})$ is defined in (\ref{T1}) and, analogously, 
\bes
\underline{\tau}^{(1)}  \doteqdot \left\{  \begin{array}{ll}
			 A^{-1}(\pi , \, a)\cdot \underline{\tau} , & \lambda_{d} < \lambda_{\pi^{-1}({d})} , \\
			A^{-1}(\pi , \, b)\cdot \underline{\tau} , & \lambda_{d} > \lambda_{\pi^{-1}({d})} . \\	
		     \end{array} \right.
\ees
The map $\hat{\R}$ preserves an invariant measure $\hat{m}$ which is the restriction to $\hat{\Upsilon}^{(1)}_{\mathscr{R}}$ of the Lebesgue measure. Denote by $p$ the projection % on $\Delta_{d-1} \times \{\pi\}\R^{d}$
\bes
p: \hat{\Upsilon}^{(1)}_{\mathscr{R}}  \rightarrow \Delta(\mathscr{R}), \qquad p  (\underline{\lambda}, \pi, \underline{\tau} )  = (\underline{\lambda}, \pi ).
\ees
The measure $p\hat{m}$ is absolutely continuous w.r.t. Lebesgue on $\Delta(\mathscr{R})$ and it is exactly the $\R$-invariant measure $\mu_{\V}$ constructed by Veech.

If $\gamma$ is a path on $\mathscr{R}$, starting at $\pi$, denote by 
\bes
\Theta(\gamma) \doteqdot \left\{ {A(\gamma)}^{-1}\, \underline{\tau}\, | \quad  \underline{\tau} \in \Theta_{\pi}  \right\} \subset \mathbb{R}^{d}.
\ees
%The observation to show that $\hat{\R}$ is the natural extension of $\R$ if the following. 
If $\gamma$ is an arrow starting at $\pi$ and ending at $\pi'$, then $\hat{\R}$ maps
\bes \left( \Delta(\gamma) \times \{\pi\} \times \Theta_{\pi}\right) \cap  \hat{\Upsilon}^{(1)}_{\mathscr{R}}  \xrightarrow{\hat{\R}} \left(\Delta_{d-1} \times \{\pi' \} \times \Theta(\gamma) \right) \cap  \hat{\Upsilon}^{(1)}_{\mathscr{R}} .
\ees
As $\underline{\lambda}$ determines the \emph{future} induction steps 
%$\hat{\R}^{n}$, $n>0$ 
(see Remark \ref{sequence}), similarly $\underline{\tau}$ determines the \emph{past} ones. %$\hat{\R}^{n}$, $n<0$. 
More precisely, let $(\underline{\lambda}^{(-i)}, \pi^{(-i)}, \underline{\tau}^{(-i)} ) \doteqdot \hat{\R}^{-i}(\underline{\lambda}, \pi, \underline{\tau} )$, for $i\in \mathbb{N}$.
\begin{rem}\label{rempast}
If $\pi'$ is the ending vertex of $\gamma=(\gamma_1,\dots, \gamma_r)$ where $\gamma_i=\gamma_i(\pi_i,c_i)$, $c_i \in \{ a,b \}$, and $\underline{\tau} \in \Theta(\gamma)$, the sequence of types and permutations obtained in the past $r$ steps of $\hat{\R}$ is determined by $\gamma$, i.e.  
%if $(\underline{\lambda}^{(-i)}, \pi^{(-i)}, \underline{\tau}^{(-i)} ) = \hat{\R}^{-i}(\underline{\lambda}, \pi, \underline{\tau} )$, for $i\in \mathbb{N}$, then  
$\RLs (\lambda^{(-i)}, \pi^{(-i)} )=A(\gamma_{r-i+1})$ and $\pi^{(-i)} = \pi_{r-i+1}$ for $i=1,\dots,r$.
\end{rem}

%$\RL{i}(p\hat(\R)^{-i} (\lambda,\pi, \tau) ) = A(\gamma_{r-i})$ for $i=1,\dots, r$.
%if $\hat{\R}^{-i}(\underline{\lambda}, \pi, \underline{\tau}) \doteqdot (\underline{\lambda}^{(-i)},\pi^{(-i)}, \underline{\tau}^{(-i)} )$ for $i=1,\dots,r$, then  
%the sequence of types and starting vertices $\pi_i$ of the arrows $\gamma_1,\dots ,\gamma_r$, i.e. $\RL{i}(T)=A(\gamma_i)$ and $\pi^{(i)} = \pi_i$. 


\subsection{Towers construction and heights vectors.}\label{towers}
The initial interval exchange $T$ can be seen as a suspension over each of the induced $T^{(r)}$ obtained at the $r^{th}$ step of Rauzy-Veech algorithm. In this subsection we define the towers which allow to retrieve $T$ from $T^{(r)}$ and $\RLp{r}$.

Note that the entries of $\RLp{r}$ have a \emph{dynamical meaning} in terms of return times. Namely, denote $I^{(r)}_j$, $1\leq j\leq d$, the subintervals of $T^{(r)}$. 
\begin{rem} \label{dynmean}
The entry $\RLp{r}_{ij}$ is equal to the number of visits of the orbit of any point $x\in I^{(r)}_j$ to the interval $I^{(0)}_i$ of the original partition before its first return in $I^{(r)}$. 
\end{rem}
Therefore, the norm $h^{(r)}_j$ of the $j^{th}$ column of $\RLp{r}$, i.e. $h^{(r)}_j \doteqdot \sum_{i=1}^{d} \RLp{r}_{ij}$ gives the return time of any $x\in I^{(r)}_j$ to $I^{(r)}$.

%As for the lengths vector, when we use the index $n$, we refer to the Zorich time $n=z_0(T)+ z_0(\Z T) + \dots z_0(\Z^{n-1} T)$. Hence
%\begin{equation} \label{norm}
%h^{(n)}_j \doteqdot \sum_{i=1}^{d} \ZLp{r}_{ij}
%\end{equation}

\paragraph{The towers.}\label{towersec}
Define 
\begin{equation}
Z^{(r)}_j \doteqdot \bigcup _{l=0}^{h^{(r)}_j-1} T^l I^{(r)}_j.
\end{equation}
When $T$ is ergodic, $\bigcup_{j=1}^{d} Z^{(r)}_j$ is a non-trivial $T$-invariant set, therefore the sets $Z^{(r)}_j$, $1\leq j\leq d$ give a partition of the whole $I$. Each $Z^{(r)}_j$ can be visualized as a tower over $I^{(r)}_j\subset I^{(r)}$, of height  $h^{(r)}_j$ (see Figure \ref{stacking}). A floor of the tower, denoted by  $Z^{(r)}_{j,l}$, is defined by
%\begin{equation}
$Z^{(r)}_{j,l} \doteqdot  T^l I^{(r)}_j$ , % \qquad 
$l=0, \dots, h^{(r)}_j-1$.
%\end{equation}
%Notice that when $n=0$, $Z^{(0)}_{j}=I^{(0)}_j$, i.e. towers of step zero are simply the subintervals of the IET.
The original $T$ is an integral map over $I^{(r)}$; under the action of $T$ every floor $Z^{(r)}_{j,l}$, but the top one ($l\neq h^{(r)}_j$), moves one step up, while $T \left( Z^{(r)}_{j,h^{(r)}_j} \right) = T^{(r)} \left( I^{(r)}_j \right)$.


\paragraph{Heights cocycle.} \label{Rauzyheights}
Let $\underline{h}^{(0)}$ be the column vector $\underline{e} \doteqdot (1, \dots , 1)^T \in \mathbb{Z}^d$ and $\underline{h}^{(n)}$ the column vector whose components are the heights $(h ^{(1)}_{1}, \dots , h ^{(1)}_{d})^T$ of the towers after the first step of the Rauzy-Veech algorithm. If we write 
%\bes 
$\underline{h}^{(1)} = R \,\, \underline{h}^{(0)}$,
%\ees
where $R=R(T)$ is a matrix in $SL(d, \mathbb{Z})$, it is easy to see that $R(T) = A(\pi,a)^T$ or $R(T) = A(\pi,b)^T$, depending on whether the Rauzy step is of type $(a)$ or $(b)$.

%%%
Hence, comparing with (\ref{lengthsrelationR}), the cocycle that determines how the vectors of the heights transform is given by the inverse transpose of the Rauzy-Veech cocycle. More precisely, if $\underline{h} ^{(r)}$ is the vector of the heights after $r$ iterations of the algorithm, then
% denoting the product of elementary Rauzy matrices along the $\R$-orbit of $T$ by
%\begin{equation} \label{productR}
%R^{(r)} = R^{(r)}(T) \doteqdot R(\R ^{r-1}(T) ) \cdot  R(\R ^{r-2}(T) )\cdot \,  \dots \, \cdot R( \R (T) ) \cdot R(T),
%\end{equation}
\be \label{hZtrasf}
\underline{h}^{(r)} = (\RLp{r})^T \, \,  
%\underline{h}^{(0)}.
\underline{e}; \qquad\underline{h}^{(s+r)} = {\RLp{r}(\R^s(T) )}^T \underline{h}^{(s)} \,  .
\ee
%The map $R$ from the space of IETs of $d$ intervals to $SL(d, \mathbb{Z})$ is a multiplicative cocycle which we will call \emph{Rauzy heights cocycle}.

%\begin{rem}\label{cocyclesrel}
%In \cite{Ve:gau, Zo:fin} there appears the definition of the cocycle $A: IETs \rightarrow SL(d, \mathbb{Z})$, such that, considering lengths vectors $\underline{\lambda}^{(r)}$ and using the same notation in formula (\ref{productR}), one has $\underline{\lambda}^{(n)}= (A^{(n)})^{-1} \underline{\lambda}^{(0)}$. It is easy to check that the heights cocycle $R$ here defined is the transpose cocycle of $A$, i.e. $R(T) = A(T)^T$ where $^T$ denotes the transpose of a matrix.
%\end{rem}

%\begin{figure}
%\centering
%\subfigure[$\lambda_{d} < \lambda_{\pi ^{-1}(d)}$]{\label{stacka}{\includegraphics[width=0.45\textwidth]{stacka.eps}}}
%\hspace{5mm}
%\subfigure[$\lambda_{d} > \lambda_{\pi ^{-1}(d)}$]{\label{stackb}{\includegraphics[width=0.45\textwidth]{stackb.eps}}}
%\caption{ Stacking for $T=(\lambda, (4321))$.}
%\label{stacking}
%\end{figure}

\subsubsection{Recurrent structure of the towers.}\label{partitionsec}
%\paragraph{Partitions of the towers.}\label{partitionsec}
%%%%Let us investigate the recursive structure of a tower $Z^{(r)}_{j_{0}}$ for given $r$ in terms of the towers $Z^{(s)}_{j}$ of previous steps $s<r$. S
%%%ELIMINABILE
\paragraph{Algorithm action on towers.}
The Rauzy-Veech algorithm can be visualized as acting on the towers, in terms of \emph{stacking} towers. One step corresponds to cutting the last tower before the permutation, i.e. $Z^{(r)}_{d}$, and stacking it over $Z^{(r)}_{\pi^{-1}d}$. In the case $a$, when ${\lambda}^{(r)}_{d} < \lambda^{(r)}_{\pi^{-1}d}$, $Z^{(r)}_{d}$ is completely cut and stacked above $Z^{(r)}_{\pi^{-1}d}$, at its right end (see Figure \ref{stacka}). In the case $b$, $\lambda^{(r)}_{d} > \lambda^{(r)}_{\pi^{-1}d}$, only the right portion of $Z^{(r)}_{d}$ of width $\lambda^{(r)}_{\pi^{-1}d}$ is cut and stacked completely above $Z^{(r)}_{\pi^{-1}d}$ (see Figure \ref{stackb}).
 
\begin{figure}
\centering
\subfigure[$\lambda_{d} < \lambda_{\pi ^{-1}(d)}$]{\label{stacka}{\includegraphics[width=0.45\textwidth]{stacka.eps}}}
\hspace{5mm}
\subfigure[$\lambda_{d} > \lambda_{\pi ^{-1}(d)}$]{\label{stackb}{\includegraphics[width=0.45\textwidth]{stackb.eps}}}
\caption{ Stacking for $T=(\lambda, (4321))$.}
\label{stacking}
\end{figure}
It is clear from the stacking description of the algorithm that each tower $Z^{(r)}_{j_{0}}$ consists of pieces of towers $Z^{(s)}_{j}$. 

\paragraph{Towers partitions.}
Define the following system of measurable partitions $\xi_s =\xi_s ( Z^{(r)}_{j_{0}} )$ of the tower $Z^{(r)}_{j_{0}}$ in terms of the subtowers $Z^{(s)}_{i}$, $0\leq s \leq r$.
The elements of the partition are complete blocks of floors of $Z^{(r)}_{j_{0}}$ which are all contained inside the same tower $Z^{(s)}_{j}$: namely, for each floor $Z^{(r)}_{j_{0},l}$ which is contained in $I^{(s)}$, construct an element $Z \in \xi_s$ in the following way. If $Z^{(r)}_{j_{0},l} \subset I^{(s)}_{j}$,  
\begin{equation}
Z \doteqdot  \bigcup_{i=0}^{h^{(s)}_{j}-1} T^i Z^{(r)}_{j_{0},l}.
\end{equation}
The set of all such $Z$ gives a partition $\xi_s$ of $Z^{(r)}_{j_{0}}$. Clearly for each  $Z \in \xi_s$ there is a unique $j$ such that $Z \subset Z^{(s)}_{j}$.
Partitions $\xi_{s'}$, $s'<s$, are refinements of $\xi_s$. 
%Indeed, $\xi_{s-1}$ is obtained from $\xi_s$ by splitting some of the elements $Z \in \xi_s$ into two elements of  $\xi_{s-1}$ (see Fig. \ref{partition}) . More precisely, if $Z$ is contained in a tower $Z^{(s)}_{j}$ not involved in the stacking at step $s-1$, it remains unchanged, while if $Z \subset Z^{(s)}_{\pi^{(s)-1}d}$ in case $b$ or $Z \subset Z^{(s)}_{\pi^{(s)-1}(d)+1}$ in case $a$, it splits into two. Notice that $\xi_0 ( Z^{(r)}_{j_{0}} )$ is the partition of the tower into floors $Z^{(r)}_{j_{0},l} $ . 

%%%%\begin{figure}
%\centering
%\includegraphics[width=1\textwidth]{partition.eps}
%\caption{An example of partitions $\xi_s ( Z^{(r)}_{j_{0}} )$ (arcs show partition elements).}
%\label{partition}
%\end{figure}

%Again, when the indices $m < n$ are used instead than $s<s$, we refer to Zorich times, i.e. $m=\sum_{i_0^{m-1}} z_0{\Z^i T}$. Zorich partitions $\{\xi_m\}_{m\in \mathbb{N}}$ are simply a subsequence of the Rauzy ones $\{ \xi_s \}_{s\in \mathbb{N}}$.

The entries of $A^{(m,n)}$ have the following meaning for the partition $\xi_m ( Z^{(n)}_{j})$.
For $m<n$, $\ZLp{m,n}_{ij}$ gives the number of visits of $x \in I^{(n)}_j$ to $I^{(m)}_i$ under the action of $T^{(m)}$ before the first return to $I^{(m)}$. Hence 
%$\ZLp{m,n}_{ij}$ gives also the number of $Z \in \xi_m ( Z^{(r)}_{j}) $ such that $C_0 \subset  Z^{(0)}_{i}=I_i$. 
\be \label{entriesBmn}
 A^{(m,n)}_{ij}= \# \{ Z \in \xi_m ( Z^{(n)}_{j}) |  \quad Z \subset Z^{(m)}_{i} \}.
\ee

\subsection{Notation Summary.}
For the convenience of the reader, we list in this section the notation relative to the induction algorithm for IETs introduced in the last paragraphs and used in the following ones.
\\
\begin{tabular}{ll}
$\R$ & Rauzy induction, $\R$: IETs $\rightarrow$ IETs; \\
$A^{-1}$ &  Rauzy lengths cocycle, $A^{-1}$: $\Delta(\mathscr{R}) \rightarrow SL(d,\mathbb{Z})$; \\
$\RL{n}$ & $ = \RLs(\R^n T)$ $n^{th}$ elementary Rauzy matrix; \\
$\RLp{n,m}$ & $=\RL{n} \dots \RL{m-1} $ product of Rauzy matrices; \\
$\RLp{n}$ & $=\RLp{0,n}$; 
\end{tabular}\\
\begin{tabular}{ll}
$I^{(n)}$ & $n^{th}$ inducing subinterval; \\
$T^{(n)}$ & IET obtained as first return on $I^{(n)}$;  \\
$I^{(n)}_j$ & $j=1, \dots , d$ subintervals exchanged by $T^{(n)}$;\\
$\lambda_j^{(n)}$ & = $|I^{(n)}_j|$ length of $I^{(n)}_j$, w.r.t. the Lebesgue measure on $I$;\\
$\underline{\lambda}^{(n)}$ &= $(\lambda_1^{(n)}, \dots, \lambda_j^{(n)}, \dots, \lambda_d^{(n)})$ lengths vector; \\
${\lambda}^{(n)}$ & = $| \underline{\lambda}^{(n)} | = \sum_{j=1}^d \lambda_j^{(n)}$ lenght $|I^{(n)}|$ of the $n^th$ inducing subinterval;
\end{tabular}\\
\begin{tabular}{ll}
$\RLp{ n ,m }_{ij }$ & number of visits of $x_0 \in I^{(m)}_{j} $ to $I^{(n)}_i$ before the first return to $ I^{(m)}$;\\
$h^{(n)}_j$ & $= \sum_i \RLp{ n ,m }_{ij }$, $j=1, \dots , d$ first return time of any $x\in I^{(n)}_j $ to $I^{(n)}$;\\
$Z^{(n)}_j$ & $= \cup_{k=1 }^{h^{(n)}_j-1} T^k  I^{(n)}_j$, for $j=1, \dots , d$ towers representing $T$ over $T^{(n)}$;\\
${h}^{(n)}$ & $= \max_{j=1,\dots, d} h^{(n)}_j$ maximum of the towers heights;\\
$\xi_{n}$ & partition into floors of step $n$, i.e $T^k(I^{(n)}_j)$, $k\leq h^{(n)}_j$, $j=1,\dots,d$;\\
$\xi_{n} (Z) $ & restriction of the partition $\xi_{n}$ to the set $Z$;
\end{tabular}\\
\begin{tabular}{ll}
$\|A\|$ & where $A\in SL(d,\mathbb{Z})$, $A\geq 0$, is given by $\|A\|=\sum_{ij} A_{ij} ; $\\
$A> 0$ & (or $A \geq0) $ if $A\in SL(d,\mathbb{Z})$ has positive (non-negative) entries; \\
$d_H$ & Hilbert distance on $\Delta_{d-1}$; \\
$diam_H$ & diameter w.r.t. the Hilbert distance; \\
$\widetilde{A}$ & $= A\underline{\lambda}/ |A\underline{\lambda}|$ projective transformation associated to $A\geq 0$;\\
$D(A)$&$= diam_h (\widetilde{A} (\Delta_{d-1}))$.
\end{tabular}\\

%%%%%%%%

\section{Growth of Birkhoff sums of derivatives% of the derivatives Birkhoff sums%of the derivatives
.}\label{growthBSsec}
%\section{Birkhoff sums of the derivatives growth% of the derivatives Birkhoff sums%of the derivatives.}\label{growthBSsec}
Let us introduce two auxiliary functions $u$, $v$ defined on $I^{(0)}$:
\bes
u(x) \doteqdot \frac{1}{ x};\qquad v(x) \doteqdot \frac{1}{1- x}.
%u(x) \doteqdot \frac{C^+}{ x};\qquad v(x) \doteqdot \frac{C^-}{1- x};
\ees
%Let $w(x)\doteqdot f'(x)- C^+ u(x) + C^- v(x)$.

\begin{prop}\label{reducetouv}
Assume $T$ is uniquely ergodic. There exists a sequence $\alpha_r$ such that $\alpha_r \rightarrow 0$ as $r \rightarrow \infty$ and for all $x$ distinct from singularities of $\BS{f}{r}$,
\bes
\BS{f'}{r}(x) = (  - C^+ + \alpha_r^+ )  \BS{u}{r}(x) + (  C^- + \alpha_r^- ) \BS{v}{r}(x), 
\ees
where $|\alpha_r^{\pm}| \leq \alpha_r$.
%For each $\epsilon > 0$ there exists $R$ such that for each $r>R$, 
\end{prop}
\begin{proof}
See Theorem 3.1 in \cite{Ko:nonI}. The same proof applies also for uniquely ergodic IETs.% using unique ergodicity.
\end{proof}

In Section \ref{growthBSusec} we prove estimates on the growth of the Birkhoff sums for $u$ and $v$ for a typical IET and then we use them in Section \ref{BSf'andf''sec} to derive some information about the growth of $\BS{f'}{r}$ and $\BS{f''}{r}$. It is enough to get estimates from $u$, since estimates from $v$ can be easily derived from the following observation.
Let $\I(x)=1-x$, %$x\in I^{(0)}$
 be the reflection on the interval $I^{(0)}$. Since $v(x)=u(\I x)$, 
%$v\cdot T^n   = u\cdot (\I \cdot T\cdot \I^{-1})^n \cdot \I$. Let us denote by $T^{\I } \doteqdot \I \cdot T \cot \I ^{-1}$. Then
%%%\bes
%\BS{v}{r}(x) = \BS{u}
$v\cdot T^n   = u\cdot (\I \cdot T\cdot \I ^{-1})^n \cdot \I  $. %\, \I I^{(0)}\rightarrow \I I^{(0)} $.  
%\ees
Let us denote by $T^{\I } \doteqdot \I \cdot T \cdot \I ^{-1}$. Hence the Birkhoff sums for $v$ with respect to $T$ and the ones for $u$ with respect to $T^{\I}$ are related by
%at $x$ can be obtained from the Birkhoff sums for $u$ with respect to $T^{\I}$ evaluated at $1-x$, or
\be\label{BSuvrel}
\BS{v,T}{r}(x)= \BS{u,T^{\I}}{r}(1-x).
\ee %, which is isometric to $I^{(0)}$. 
Remark that if $T=((\lambda_1,\lambda_2, \dots, \lambda_n),\pi)$, then $T^{\I}= ((\lambda_n,\lambda_{n-1}, \dots, \lambda_1),\pi^{\I})$ where $\pi^{\I}\doteqdot (n \, n\!-\!1\dots 2\, 1)\cdot \pi \cdot (n\, n\!-\!1\dots 2\, 1)$. Hence the map $T\mapsto T^{\I}$ from $\Delta_{d-1} \times \mathscr{R}(\pi) \rightarrow \Delta_{d-1} \times \mathscr{R}(\pi^{\I})$ preserves the Lebesgue measure.

%For Property $3$, let $\overline{l}$ is the maximum lenghts of the paths $\gamma_{s_i}$. Since $n_{l+\overline{l}}\geq n_{l}+\overline{l}$, $\RLs{n_l, n_{l+\overline{l}}}$ by Remark )

%\begin{proposition}\label{uvasymptotics}
%For each $\nu> 1$, there is a set of IETs with full measure such that when considering the suspension flow with roof $u$ and $v$ respectively and base $T$ in this set, there exists $T>0$ and a sequence of partial partition $\xi_t$, $Leb(\xi_t)\rightarrow 0$ as $t \rightarrow \infty$, such that for each $t>T$, $x \in \xi_t$ and $r=r(x,t)$,  
%\begin{eqnarray}
%r \log (1-\epsilon) \leq \BS{u}{r} \leq (1+ \epsilon) r \log r \\
%r \log (1-\epsilon) \leq \BS{v}{r} \leq (1+ \epsilon) r \log r .
%\end{eqnarray}
%\end{proposition}%

%Assume that a certain asymmetry $C+>C^-$ is given. It is then possible to choose $\epsilon >0$ so that $C^+(1-\epsilon)^2/C^- (1+ \epsilon)^2 > 1$. Assume $T$ is uniquely ergodic and it also belongs to the full measure set in Proposition \ref{uvasymptotics}. Such assumption is full measure by Veech result. Choose $T>0$


\subsection{A diophantine-type condition for IETs.} \label{existencebalancedtimessec}
In this section we define the set of full measure of IETs for which we prove Theorem \ref{mixing}.
%give the precise definition of the typical condition we require on the interval exchange $T$.
Proposition \ref{existencebalancedtimes} shows that for typical $T$ one can find a subsequence $\{ n_l\}_{l\in \mathbb{N}}$ of induction times such that the corresponding IETs $\{ \R^{n_l} T \}_{l\in \mathbb{N}}$ in the Rauzy orbit $\{ \R ^n T \}_{n\in \mathbb{N}}$ enjoy some good properties (listed in the Proposition \ref{existencebalancedtimes} below), which we call \emph{balance}; moreover it gives a control on their frequencies.  
%Balance guarantees in particular that the subintervals $I^{n_l}_j$ of the IETs $\Z^{n_l} T $ have approximately the same lengths. 
In the next Sections we will use these balanced induction times 
% Section \ref{Birkhoffgrowth}
in order to estimate the growth of $\BS{u}{r}$.   

%%%%%ELIMINABILE
Balanced times are related to occurrences of some positive matrices in the renormalization cocycle. Conditions on the frequencies of occurrence of such matrices play for interval exchanges a role analogous to diophantine conditions for rotations. Different type of estimates in this spirit appear in the works of \cite{Ke:sim, MMY:coh, Bu:dec, AGY:exp}. The condition that we use is derived from \cite{AGY:exp}.

%\subsubsection{Preliminaries.}
%We need to remind the following definitions to state Proposition \ref{existencebalancedtimes}.
%\paragraph{Hilbert metric and projective contractions.}
%Consider on the simplex $\Delta_{d-1} \subset \mathbb{R}_+^d$ the \emph{Hilbert distance} $d_H$, defined as follows.
%\bes
%d_H(\lambda, \lambda') \doteqdot \log \frac{\max_{i=1,\dots , d}\frac{\lambda_i}{\lambda'_i}}{\min_{i=1 , \dots ,d}\frac{\lambda_i}{\lambda'_i}}.
%\ees
%We denote the diameter with respect to $d_H$ of a projective subset $\Lambda \subset \Delta_{d-1}$ by
%\be \label{contraction}
%diam_H (\Lambda ) \doteqdot \sup_{\lambda, \lambda' \in \Lambda} d_H( \lambda, \lambda' ) .
%\ee
%Remark that if its closure $\overline{\Lambda} \subset \Delta_{d-1}$, 
%  then $diam_H (\Lambda )$ is finite. 

%Let us recall that to each matrix $A\in SL(d,\mathbb{Z})$ one can associate a projective transformation $\widetilde{A}: \Delta_{d-1} \rightarrow \Delta_{d-1}$ given by
%\bes
%\widetilde{A} \lambda = \frac{A \lambda }{|A \lambda |}.
%\ees
%If $A \geq 0$, $d_H(\widetilde{A} \lambda, \widetilde{A} \lambda' ) \leq d_H(\lambda, \lambda')$. If furthermore $A>0$, we get a definite contraction. More precisely, $A>0$ is equivalent to $\widetilde{A} \left( \overline{\Delta}_{d-1} \right) \subset \Delta_{d-1}$, where  and hence defining 
%\be \label{D_Adef}
%D_A \doteqdot diam_H \left( \widetilde{A} \left( \overline{\Delta}_{d-1} \right) \right) , 
%\ee
%we have $D_A < \infty$. Then
%\be \label{contraction}
% d_H(\widetilde{A} \lambda, \widetilde{A} \lambda' ) \leq (1-e^{-D_A}) d_H(\lambda, \lambda').
%\ee

\subsubsection{Existence of balanced return times.}%\label{existencebalancedtimessec}
\begin{prop}\label{existencebalancedtimes}
Let $1<\tau <2$. For each $\pi$ and for Lebesgue a.e. $\lambda\in  \Delta_{d-1}$, there exist a subsequence $\{ n_l\}_{l\in \mathbb{N}}$ of induction times, $\nu>1$, $\kappa >1$, $0<D < \infty$ and $\overline{l} \in \mathbb{N}$, such that the following properties hold for all $l\in \mathbb{N}$:
%%%%ELIMINABILI Titoletti (o metterli assieme)
\begin{enumerate}
\item \emph{( $\nu$ -balance of  lengths )}
\be \label{lengthsbalance} 
\frac{1}{\nu} \leq \frac{\lambda_i^{(n_l)}}{\lambda_j^{(n_l)}} \leq \nu, \quad  \forall \, 1 \leq i, \, j \leq d;  
% \qquad \mathrm{(\nu -balance \, of \, lengths )}
\ee
\item \emph{($\kappa$ -balance of  heights)}
\be \label{heightsbalance} 
\frac{1}{\kappa} \leq \frac{h_i^{(n_l)}}{h_j^{(n_l)}} \leq \kappa, \quad  \forall \, 1 \leq i, \, j \leq d;  
%\qquad \mathrm{(\kappa -balance \, of \, heights)}
\ee
\item \emph{(positivity)} %For all $l\in \mathbb{N}$,\footnote{Recall that $D(A)$ was defined in (\ref{D_Adef}).}
\be \label{positive} 
\ZLp{n_{l\phantom{\overline{l}}},n_{l+\overline{l}} }> 0, \quad \mathrm{and}\footnote{Recall that $D(A)$ was defined in (\ref{D_Adef}).}\quad D({\ZLp{n_{l\phantom{\overline{l}}},n_{l+\overline{l}}}}) \leq D .  
%\qquad \mathrm{(positivity)}
\ee
\item \emph{(integrability)}
\be \label{integrability} 
\lim _{l \rightarrow + \infty} \frac {\|\ZLp{n_l,n_{l+1}} \|}{l^{\tau}} =0 .
%\qquad \mathrm{(integrability)} .
\ee
%old condition \be \label{integrability} 
%\frac{\log \|\ZLp{n_l,n_{l+1}}\|}{n_l} \xrightarrow{l \rightarrow \infty} 0   \qquad \mathrm{(integrability)} .
%\ee
\end{enumerate}
\end{prop}

%\begin{prop}\label{existencebalancedtimes}
%There exists a set $G \subset \Delta_{d-1} \times \R (\pi)$ of IETs with full measure, i.e. $\mu_{\Z}(G)=1$, such that for any $T \in G$ and for any $\nu > 1$ there is $\kappa>0$ and there exists a subsequence $\{ n_l\}_{l\in \mathbb{N}}$ of Zorich times such that the following properties hold:
%For any $\nu > 1$ and any $\pi \in \Sym{d}$, there exists a set $\Delta_{\nu} \subset \Delta$ of  of full measure such that for any IET $(\lambda, \pi)=T$ with $\lambda \in \Delta_{\nu} $, there exist a subsequence $\{ n_l\}_{l\in \mathbb{N}}$ of Zorich times such that the following properties hold:
%\begin{itemize}
%\be \label{lengthsbalance} 
%\frac{1}{\nu} \leq \frac{\lambda_i^{(n_l)}}{\lambda_j^{(n_l)}} \leq \nu, \quad \forall i,j=1,\dots, d;   \qquad \mathrm{(\nu -balance \, of \, lengths )}
%\ee
%\be \label{heightsbalance} 
%\frac{1}{\nu} \leq \frac{h_i^{(n_l)}}{h_j^{(n_l)}} \leq \kappa, \quad \forall i,j=1,\dots, d;  \qquad \mathrm{(\kappa -balance \, of \, heights)}
%\ee
%\be \label{positive} 
%\ZLp{n_l,n_{l+1}} > 0, \quad \forall l\in \mathbb{N};  \qquad \mathrm{(positivity)}
%\ee
%\be \label{controlnusB}
%\nu_{row}(\ZLp{n_l,n_{l+1}} ) \leq \nu ;\qquad 
%\nu_{col}(\ZLp{n_l,n_{l+1}} ) \leq \kappa ;
%\ee
%\be \label{integrability} 
%\%frac{\log \|\ZLp{n_l,n_{l+1}}\|}{l} \xrightarrow{l \rightarrow \infty} 0   \qquad \mathrm{(integrability)} .
%\ee
%old condition \be \label{integrability} 
%\frac{\log \|\ZLp{n_l,n_{l+1}}\|}{n_l} \xrightarrow{l \rightarrow \infty} 0   \qquad \mathrm{(integrability)} .
%\ee
%\end{prop}
%\begin{def}\label{balancedtimedef}
A return time which satisfies Properties $1$ and $2$  will be called \emph{balanced return time}. 
%\end{def}
A balanced return time is such that 
%\ref{lengthsbalance} and \ref{heightsbalance} guarantees that 
the lengths and the heights of the induction towers
%of the subintervals of the induced IET and the first return times correspoding to heights of Rohlin towers 
are approximately of the same size. 
%Condition (\ref{positive}) 
Property $3$ gives some uniform distribution of subintervals of time $n_{l+\overline{l}}$ inside the subintervals of the previous balanced time $n_l$.
%The sequence $\{ n_l\}_{l\in \mathbb{N}}$ will be defined as a translate $ n_l = \widetilde{n_l} + a$, $a\in \mathbb{N}$, of the sequence $\{ \widetilde{n{_l}} \}_{l\in \mathbb{N}}$ given by return times of $\Z^n T$ on a subsimplex $\Delta' \subset \Delta_{d-1}$, i.e. defined inductively by:
%\bes
%\widetilde{n_l} = \min \left\{ n \, |\quad  n > \widetilde{n}_{l-1} \, \, \mathrm{and}\, \, \Z ^n T \in \Delta' \right\}.
%\ees
Property $4$ % (\ref{integrability})
 is the diophantine condition which guarantees some control of the frequencies of occurrence of balanced times. It will be deduced from the power integrability of a certain induced cocycle, proved in \cite{AGY:exp}. 

We remark that (\ref{integrability}) is analogous to the diophantine condition used for rotations in \cite{SK:mix}, i.e. $k_l = o( l^{\tau})$, where $\{ k_l \}_{l \in \mathbb{N}}$ are the entries of the continued fraction and the exponent $\tau$ satisfies the same assumption $1 < \tau <2$.  
%This will allow us to deduce property (\ref{integrability}) from the integrability of Zorich cocycle. 

%%%ELIMINABILE defn
\begin{defn}\label{Mdef}
Let $\M^+ = \M^+ (\Delta_{d-1}\times \mathscr{R}(\pi))$ be the set of IETs in $\Delta_{d-1}\times \mathscr{R}(\pi)$ such that Proposition \ref{existencebalancedtimes} hold and $\M^- = \M^- ( \Delta_{d-1} \times \mathscr{R}(\pi) )$ be the set of $T\in \Delta_{d-1} \times \mathscr{R}(\pi)$ such that $T^{\I} \in \M^+ (\Delta_{d-1} \times \mathscr{R}(\pi^{\I}))$.
Denote $\M = \M^+ \cap \M^-.$
\end{defn}
The IETs in $\M$ are the ones for which we prove mixing of the suspension flows having one asymmetric logarithmic singularity.
\begin{rem} The set $\M$ has full measure. %, since the map $T \rightarrow T^{\I}$ preserves the Lebesgue measure. 
Indeed, $\M^+$ has full measure by Proposition \ref{existencebalancedtimes} and also $\M^-$ has full measure since, as already remarked, $T \mapsto T^{\I}$ preserves the Lebesgue measure.
\end{rem}
\begin{rem}
The IETs in $\M$ are uniquely ergodic, as it follows from Property (\ref{positive}) in Proposition \ref{existencebalancedtimes} with the help of techniques used by Veech in \cite{Ve:gau, Ve:pro}.
\end{rem}

%The remark follows by noticing that the map $T \mapsto T^{\I}$ preserves the Lebesgue measure.

%The IETs in $\M$ are the ones for which we prove mixing of the suspension flows.

%%%%ELIMINABILE
In the remaining part of this section we derive Proposition \ref{existencebalancedtimes} from a result in \cite{AGY:exp} and then prove some simple corollaries, which will be used later.

%\subsubsection{Preliminaries.}
%\paragraph{Hilbert metric and projective contractions.}
%Consider on the simplex $\Delta_{d-1}$ the \emph{Hilbert metric} $d_H$, defined as follows.
%\begin{equation}
%d_H(\lambda, \lambda') \doteqdot \log \frac{\max_{i=1,\dots , d}\frac{\lambda_i}{\lambda'_i}}{\min_{i=1 , \dots ,d}\frac{\lambda_i}{\lambda'_i}}.
%\end{equation}
%We denote the diameter with respect to $d_H$ of a $\Lambda \subset \Delta_{d-1}$ by
%\be \label{contraction}
%diam_H (\Lambda ) \doteqdot \sup_{\lambda, \lambda' \in \Lambda} d_H( \lambda, \lambda' ) .
%\ee
%Remark that if $\Lambda$ is compact, $diam_H (\Lambda )$ is finite. %

%Let us recall that to each matrix $A\in SL(d,\mathbb{Z})$ one can associate a projective trasformation $\widetilde{A}: \Delta_{d-1} \rightarrow \Delta_{d-1}$ given by
%\bes
%\widetilde{A} \lambda = \frac{A \lambda }{|A \lambda |}.
%\ees
%If $A \geq 0$, $d_H(\widetilde{A} \lambda, \widetilde{A} \lambda' ) \leq d_H(\lambda, \lambda')$. If furthermore $A>0$, we get a definite contraction. More precisely, $A>0$ is equivalent to the closure $\widetilde{A} \left( \overline{\Delta}_{d-1} \right) \subset \Delta_{d-1} $ and hence defining 
%\be \label{D_Adef}
%D_A \doteqdot diam_H \left( \widetilde{A} \left( \overline{\Delta}_{d-1} \right) \right) , 
%\ee
%we have $D_A < \infty$. Then
%\be \label{contraction}
% d_H(\widetilde{A} \lambda, \widetilde{A} \lambda' ) \leq (1-e^{-D_A}) d_H(\lambda, \lambda').
%\ee

%\paragraph{Control of balance.}
%It is easy to see that $\nu_{row}(B)$ (respectively $\nu_{col}(B)$) controls the ratio
%of lengths and heights respectively, i.e.
% of norms of rows (columns) of $B$, i.e. 
%\begin{eqnarray}
%\frac{1}{\nu_{row}(B)} &\leq&  \frac{\sum_j B_{ij }} {\sum_j B_{lj}} \leq  \nu_{row}(B)  \qquad \forall i,j,l , \label{row} \label{rowcolcontrol} \\
%\frac{1}{\nu_{col}(B)} &\leq& \frac{\sum_i B_{ij }} {\sum_i B_{ik}} \leq  \nu_{col}(B)  \qquad \forall i,j,k , \label{column}
%\end{eqnarray}
%and that if $A$ has non-negative entries and non-zero determinant,
%\be 
%\nu_{row}(BA) \leq \nu_{row}(B) , \qquad \nu_{col}(AB) \leq \nu_{col}(B). \label{colprop}
%\ee
%Remark also that $\nu_{row}(B) = \nu_{col}(B^T)$, where $B^T$ is the transpose of $B$.
%One can easily prove the following.
%\begin{lemma}
%If $\underline{\lambda} \in B \Delta_{d-1}$, 
%\bes
%\frac{1}{\nu_{row}(B)} \leq \frac{\lambda_i}{\lambda_j} \leq \nu_{row} (B) , \quad \forall i,j=1,\dots, d.
%\ees
%If $\underline{h} \in B^T \Delta_{d-1}$,
%\bes
%\frac{1}{\nu_{col}(B)} \leq \frac{h_i}{h_j} \leq \nu_{col} (B) , \quad \forall i,j=1,\dots, d.
%\ees
%\end{lemma}


%\paragraph{Integrability of induced cocycles.}
%The following general fact can be proven using Birkhoff ergodic theorem.
%\begin{prop}\label{inducedintegral}
%Let $\mu$ be a probability measure on $X$, $T: X \rightarrow X$ be a $\mu$-preserving and ergodic transformation and let $f \in L_1(X,\mu)$ be an $\mu$-integrable function. Let $Y \subset X$ and for each $y\in Y$ consider the first return time $r(y)$, minimum $r\in \mathbb{N}$ such that $T^n(y) \in Y$. Then 
%\be
%\int_Y   \sum_{i=0}^{r(y)-1} f(T^i y) \ud \mu = \int_X f(x) \ud \mu .
%\ee
%\end{prop}

%In particular, we get the following corollary. Let $\Delta' \subset \Delta $ and given $T=(\lambda ,\pi)$ with $\lambda \in \Delta'$,  define the induced cocycle $\ZLs_{\Delta'}$ by
%\bes
%\ZLs_{\Delta'}(T) \doteqdot  \ZLp{r(T)}(T) ,
%\ees
%where $r(T)$ is the first return time of $T$ to  $\Delta' \times \{\pi\}$.
%\begin{cor}\label{intDelta'}
%For any $\Delta' \subset \Delta $, the induced cocycle $\ZLs_{\Delta'}$ is also integrable, i.e.
%\bes
%\int_{\Delta'} \log \|\ZLs_{\Delta'}{(T)} \| \ud \mu(T) < + \infty.
%\ees  
%\end{cor}
%\begin{proof} 
%Apply Proposition \ref{inducedintegral} taking $X$ the space of IETs and as $\mu$ the invariant probability measure $\mu_{\Z}$ constructed by Zorich and considering as $T$ the ergodic transformation $\Z$. Let $Y = \Delta' \times \R (\pi)$. Remarking that, if $r(T)$ is the first return time of $T$ to $Y$ under the action of $\Z$,  
%\be
%\log \| \ZLp{r(T)}(T) \| \leq  \sum_{i=0}^{r(T)-1} \log \|  \ZLs ( Z^i T ) \| 
%\ee
%and using the integrability of the Zorich cocyle (see Theorem. \ref{Zorich}), we get the corollary.
%\end{proof}

%Using a standard Borel-Cantelli argument, from Corollary \ref{intDelta'} we also get
%\begin{cor}\label{corintDelta'}
%For $\mu_{\Z}$-a.e. $\lambda \in \Delta'$,
%\be
%\lim_{l\rightarrow \infty} \frac{ \log \| \ZLs_{\Delta'}^{(l)}{(T)} \|}{l} =0. 
%\ee
%\end{cor}

\subsubsection{Proof of Proposition \ref{existencebalancedtimes}.}
%Let us recall the following result, proved in \cite{AGY:exp}. % Theorem $4.10$.% in \cite{AGY:exp}.
If $Y\subset \hat{\Upsilon}^{(1)}_{\mathscr{\R}}$, let $\RLs _{Y}$ denote the induced cocycle of the Rauzy-Veech lengths cocycle associated to first returns to $Y$ under $\hat{\R}$, i.e. for $(\underline{\lambda}, \pi, \underline{\tau} )\in {Y}$,
\bes
\RLs_{Y}(\underline{\lambda}, \pi, \underline{\tau}  ) \doteqdot \RLp{r_{Y}} \left( (\underline{\lambda}, \pi ) \right), \quad \mathrm{where} \quad r_{Y}\doteqdot \min \{ r\in \mathbb{N}^+| \quad \hat{\R}^r (\underline{\lambda}, \pi, \underline{\tau}  ) \in Y \}.
\ees
%Remark that $r_Y$ depends on $(\underline{\lambda}, \pi)$ only and not on $\underline{\tau}$.

%%%ELIMINABILE
The following result is proved in \cite{AGY:exp}.
\begin{thm}[Avila, Gou\"{e}zel, Yoccoz]\label{AGY}
For every $\delta>0$ there exists a finite union
\bes
\hat{Z}^{(1)}
 \doteqdot \left( \bigcup_{i=1}^{n} \Delta(\gamma_{s_i}) \times \{ \pi_i \} \times \Theta(\gamma_{e_i})\right) \cap \hat{\Upsilon}^{(1)}_{\mathscr{R}}, 
\ees
where $\pi_i$ is both the initial permutation of the path $\gamma_{s_i}$ and the final one of the path $\gamma_{e_i}$ and where $A(\gamma_{s_i})>0$ and $A(\gamma_{e_i})>0$ for all $i=1, \dots, n$, such that
%, denoting by $\ZLs_{\hat{Z}^{(1)}}$ is the induced cocycle given by first returns to $\hat{Z}^{(1)}$, i.e. for $(\underline{\lambda}, \pi, \underline{\tau} )\in \hat{Z}^{(1)}$,
%\bes
%\ZLs_{\hat{Z}^{(1)}}(\underline{\lambda}, \pi, \underline{\tau}  ) \doteqdot \ZLp{r_{\hat{Z}^{(1)}}}(\lambda, \pi ), \quad \mathrm{where} \quad r_{\hat{Z}^{(1)}}\doteqdot \min \{ r\in \mathbb{N}| \quad \hat{\R}^r (\underline{\lambda}, \pi, \underline{\tau}  ) \in \hat{Z}^{(1)} \}.
%\ees
\be\label{integrabilityAGY}
\int_{\hat{Z}^{(1)}
} \| \RLs_{\hat{Z}^{(1)}
}  \|^{1-\delta} \ud \hat{m} < \infty.
\ee
\end{thm}
%A few comments about the statement.
\noindent Theorem \ref{AGY} is a reformulation of Theorem $4.10$ in \cite{AGY:exp}. The original statement claims the integrability of $e^{(1-\delta)r_{\hat{Z}^{(1)}}}$, where $r_{\hat{Z}^{(1)}}$ is the first return time of $(\underline{\lambda} , \pi, \underline{\tau}) \in \hat{Z}^{(1)}$ under the Veech flow, which is given by
%%%
\bes
r_{\hat{Z}^{(1)}}( (\underline{\lambda} , \pi, \underline{\tau})) \doteqdot  - \log | \RLs_{\hat{Z}^{(1)}%^{(1)}
}^{-1} \, \underline{\lambda} \, | =  %\log \left| \frac{\lambda}{ \RLs_{\hat{Z}^{(1)}^{(1)}}^{-1} \, \underline{\lambda} \, }\right|
  \log | \RLs_{\hat{Z}^{(1)}%^{(1)}
}\, \underline{\lambda }'  \, |, 
\ees
where $(\underline{\lambda}',\pi')= R^{r_{\hat{Z}^{(1)}}} (\underline{\lambda} , \pi)$. The second equality follows by taking norms of $\RLs_{\hat{Z}^{(1)}}\, \underline{\lambda }' = \underline{\lambda } /   | \RLs_{\hat{Z}^{(1)}}^{-1} \, \underline{\lambda} \, |$. Since $\underline{\lambda }'$ belong to the compact set $\cup_{i=1}^{n} \Delta(\gamma_{s_i}) $, 
\bes
 \log | \RLs_{\hat{Z}^{(1)}%^{(1)}
} \underline{\lambda }'  | \geq   \log ( \min_i\lambda'_i \,  \| \RLs_{\hat{Z}^{(1)}%^{(1)}}
 } \| ) \geq const + \log \| \RLs_{\hat{Z}^{(1)}%^{(1)}
}\| .
%const  \| \RLs_{\hat{Z}%^{(1)}
%}  \| \leq \log ( \min_i\lambda'_i ) \| \RLs_{\hat{Z}^{(1)}}  \|  \log \| \RLs_{\hat{Z}%^{(1)}
%} \underline{\lambda ' } | \leq \log ( \max_i\lambda'_i ) \| \RLs_{\hat{Z}%^{(1)}
%}  | \leq const'  \| \RLs_{\hat{Z}^{(1)}}  \|.
\ees 
Hence (\ref{integrabilityAGY}) follows from the integrability of $e^{(1-\delta)r_{\hat{Z}^{(1)}}}$.
Positivity of  $A(\gamma_{s_i})$ and $A(\gamma_{e_i})$ is clear from the proof of Thm. 4.10, in which $\gamma_{s_i}$ and  $\gamma_{e_i}$ are chosen minimal and $2d-3$ complete and hence positive by Lemma $3.3$.  

%%%%ELIMINABILE Maybe to remove??
\begin{proofof}{Proposition}{existencebalancedtimes}
Given $1<\tau<2$, let $\delta\doteqdot 1-\tau^{-1}>0$. Let $\hat{Z}^{(1)}$ be the corresponding set given by Theorem \ref{AGY}. Let $\overline{l}$ be the maximum length of the paths $\gamma_{s_i}$ and $\gamma_{e_i}$ for $i=1,\dots, n$.

Given $(\underline{\lambda},\pi)$, choose any $\underline{\tau} \in \Theta_{\pi}$. Let $\{n_l \}_l\in \mathbb{N}$ be the subsequence of visits of the $\hat{\R}$ orbit of  $(\underline{\lambda},\pi , \underline{\tau})$ to $\hat{Z}^{(1)}$ given by
\begin{eqnarray}\label{sequencedef}
n_0 &\doteqdot &\min \{ n\in \mathbb{N}^+ |\, n\geq \overline{l}, \,  \hat{\R}^n (\underline{\lambda},\pi , \underline{\tau}) \in \hat{Z}^{(1)} \}; \\
n_{l+1} &\doteqdot &\min \{ n \in \mathbb{N}^+ |\, n> n_l , \, \hat{\R}^n (\underline{\lambda},\pi , \underline{\tau}) \in \hat{Z}^{(1)} \}; \qquad l \in \mathbb{N^+}. 
\end{eqnarray}
Notice that (\ref{sequencedef}) is independent on $\underline{\tau}$, since as soon as $n$ is bigger than the maximum length $\overline{l}$ of the paths $\gamma_{e_i}$, visits to $\hat{Z}^{(1)}$ are determined by $\underline{\lambda}$ only (see Remark \ref{sequence} and \ref{rempast}).
%\ees
%Remark that $\{n_l \}_l\in \mathbb{N}$ depends on  $(\underline{\lambda},\pi)$ only (and not on $\tau$).
Let us show that Properties $1$, $2$ (balance) and $3$ (positivity) of Proposition \ref{existencebalancedtimes} automatically hold for $(\underline{\lambda}, \pi)$ and the sequence  $\{n_l \}_{l\in \mathbb{N}}$. 

Since  by definition $\hat{\R}^{n_l} (\underline{\lambda},\pi, \underline{\tau}) \in \Delta(\gamma_{s_j})\times \{ \pi_j \} \times \Theta ( {\gamma_{s_j}})  $ for some $j$, in particular $\lambda^{(n_l)} / | \lambda^{(n_l)}| \in  \Delta(\gamma_{s_j}) $. By positivity of the $A( \gamma_{s_i} ) >0$, the union $\cup_i \Delta(\gamma_{s_i}) $ is compact and hence (see subsection \ref{Hilbertsec}) contained in a ball for the Hilbert metric $d_H$, centered at $(1/d, \dots, 1/d)$, of some radius $r_{s}>0$. Hence, 
%$d_H(\lambda^{(n_l)} / | \lambda^{(n_l)}| ,(1/d, \dots, 1/d  ) \leq \epsilon_s$ or $\min_i \lambda^{(n_l)}_i / \max_i \lambda^{(n_l)}_i \leq e^{\epsilon_s}$, 
\bes
d_H \left( \frac{\underline{\lambda}^{(n_l)} }{ | \underline{\lambda}^{(n_l)}|}  ,\left( \frac{1}{d}, \dots, \frac{1}{d}  \right) \right) \leq r_s \quad \mathrm{or}\, \mathrm{equivalently} \quad \frac{  \max_i \lambda^{(n_l)}_i }{ \min_i \lambda^{(n_l)}_i } \leq e^{r_s},
\ees
which, setting $\nu\doteqdot e^{r_s}> 1$,  is $\nu$-balance of lengths.

Similarly, from $\underline{\tau}^{(n_l)} \in \Theta ( {\gamma_{e_j}})$ we get by Remark \ref{rempast} that $\underline{\lambda}^{(n_l)} = A(\gamma_{e_j})^{-1} \underline{\lambda}^{(n_l-L)}$, where $L$ is length of $\gamma_{e_j}$. Since the heights transform according to (\ref{hZtrasf}), $\underline{h}^{(n_l)} = A(\gamma_{e_j})^T \underline{h}^{(n_l-L)}$. Arguing as above, by compactness, the union $\cup_i \widetilde{A(\gamma_{e_i})^T} \Delta_{d-1}$ is contained in a ball centered at $(1/d, \dots, 1/d)$ of some radius $r_{e}>0$ and this gives $\kappa\doteqdot e^{r_e}$ balance of the heights.

For Property $3$, %let $\overline{l}$ is the maximum lengths of the paths $\gamma_{s_i}$. S
since $\overline{l}$ is the maximum lengths of the paths $\gamma_{s_i}$ and $n_{l+\overline{l}}\geq n_{l}+\overline{l}$, by Remark \ref{sequence} we have $\RLp{n_l, n_{l+\overline{l}}} = A(\gamma_{s_j}) A $ for some $A\geq 0$. Hence $\RLp{n_l, n_{l+\overline{l}}}>0$ and $D(\RLp{n_l, n_{l+\overline{l}}}) \leq D\left( A(\gamma_{s_j}) \right) \leq 2 r_s$.

Let us show first that Property $4$ holds for typical $(\underline{\lambda},\pi,\underline{\tau})\in \hat{Z}^{(1)}$. Remark that $ \RLp{n_l, n_{l+1}} ( \underline{\lambda}, \pi, \underline{\tau } ) = \RLs_{\hat{Z}^{(1)}} (\hat{R}^{n_l} (\underline{\lambda},\pi, \underline{\tau}) ) $.
For each $\epsilon_k\doteqdot 1/k$, by $\hat{\R}$ invariance of $\hat{m}$, 
\bes
%\hat{m} \{ \| \RLp{n_l, n_{l+1}} \|  \geq \epsilon_k l^{\tau} \} = \hat{m} \{ \|  \RLs_{\hat{Z}^{(1)}}   \|  \geq \epsilon_k l^{\tau} \} =  \hat{m} \{ \|  \RLs_{\hat{Z}^{(1)}}  \| ^{\tau^{-1}} \epsilon_k ^{-\tau^{-1}} \geq l  \} .
\begin{split}
\hat{m} \{ ( \underline{\lambda}, \pi, \underline{\tau } ) \in \hat{R}^{n_l} | &\,\quad \| \RLp{n_l, n_{l+1}}( \underline{\lambda}, \pi, \underline{\tau } )  \|  \geq \epsilon_k l^{\tau} \}% = \hat{m} \{ \|  \RLs_{\hat{Z}^{(1)}}   \|  \geq \epsilon_k l^{\tau} \}
= \\ &=  \hat{m} \{  ( \underline{\lambda}, \pi, \underline{\tau } ) \in \hat{R}^{n_l} | \quad \|  \RLs_{\hat{Z}^{(1)}} ( \underline{\lambda}, \pi, \underline{\tau } )   \| ^{\tau^{-1}} \epsilon_k ^{-\tau^{-1}} \geq l  \} .
\end{split}
\ees
Since we chose $\tau^{-1}=1-\delta$, the integrability condition (\ref{integrabilityAGY}) implies that $\sum_l  \hat{m} \{ \|  \RLs_{\hat{Z}^{(1)}}  \| ^{\tau^{-1}}  \epsilon_k ^{-\tau^{-1}} \geq l  \} < \infty$ for each $\epsilon_k$. Hence, it follows by Borel-Cantelli that there exists a subset $\hat{Z}'\subset \hat{Z}^{(1)}$ with $\hat{m}(\hat{Z}')= \hat{m} (\hat{Z}^{(1)})$ such that for $(\underline{\lambda}, \pi, \underline{\tau })\in \hat{Z}'$ the sequence $n_l$ satisfies (\ref{integrability}). 

Consider the projection $p\hat{Z}'\subset \Delta(\mathscr{R})$.
%Notice that (\ref{integrability}) is independent on $\underline{\tau}$, since as soon as $n$ is bigger than the maximum length of the paths $\gamma_{e_i}$, visits to $\hat{Z}$ are determined by $\underline{\lambda}$ only (see Remark \ref{sequence} and \ref{rempast}). Hence, 
By indipendence of the definition of $\{n_l\}$ on $\tau$ , for each $(\underline{\lambda},\pi ) \in p\hat{Z}'$ we have (\ref{integrability}). Moreover, $p\hat{Z}'$ has $p\hat{m}$ full measure in $p \hat{Z}^{(1)}= \cup_i \Delta(\gamma_{s_i})\times \{ \pi_i \}$ and in particular positive $p\hat{m}=\mu_{\V}$ measure and hence also $\mu_{Z}$ positive measure. 

To conclude, let $\M^+$ be the set of $T\in \Delta(\mathscr{R})$ such that there exists $\overline{n}$, for which $R^{\overline{n}}T\in p\hat{Z}'$. Clearly if $T\in \M^+$, all the Properties are satisfied by the sequence $n_l \doteqdot \widetilde{n}_{l}+\overline{n}$, where $\widetilde{n}_l$ is the sequence associated to $\R^{\overline{n}}T$. To see that $\M^+$ has full measure, it is enough to use ergodicity of $\Z$ and the fact that $\mu_{\Z} (p\hat{Z}' )>0$, remarking  that $\Z$ orbits are subsets of $\R$ orbits. The formulation in Proposition \ref{existencebalancedtimes} follows by absolute continuity of $\mu_{\Z}$ w.r.t. Lebesgue. %By absolute continuity of $\mu_{\Z}$, $\mu{\Z}(\M^+)=1$ is equivalent to for each $\pi, $Lebesgue a.e. $\underline{lambda}$ is such that $(\underline{\lambda}, \pi)\in \M^+$.


\end{proofof}


%\paragraph{Definition of the balanced subsimplex $\Delta_{\nu}$.}
%Fix $\nu >1$ and $\pi' \in \R(\pi)$. 
%Consider $B_d (\underline{1}, \log \nu /2 ) \subset \Delta_{d-1}$, ball of radius $\log \nu /2 $ with respect to $d_H$, centered at $\underline{1}=(1,\dots,1)\in \Delta_{d-1}$. Let $\underline{\lambda}' \in B_d (\underline{1}, \log \nu /2 ) $ be such that $T'=(\underline{\lambda}',\pi')$ is minimal, uniquely ergodic and a Birkhoff generic point for $\Z$. The existence of such a $\underline{\lambda}'$ follows from the Theorems \ref{Veechergodic} and  \ref{Veechuniquelyergodic} by Veech and from a result by Keane in \cite{Ke:int}.%

%Consider the subsequence $\{ n_k \}_{k\in \mathbb{N}}$ of Zorich times given by return times of the orbit $\{ \Z^n{T'}\} _{n\in \mathbb{N}}$  to $\Delta_{d-1} \times \{ \pi' \}$. The corresponding IETs, $\Z^{n_k}{T}$, share the same combinatorics, i.e. they are of the form $(\underline{\lambda'}^{(n)}/\lambda'^{(n)} , \pi')$. 

%Recall that $\ZLp{n}(T')$ denotes products of matrices in the Zorich expansion of $T'$.
%Since $T'$ is minimal, there exist $k_0$ such that $\ZLp{n_{k_0}}(T')>0$. Denote by 
%\bes
%\Delta(\ZLp{n}(T') ) = \left\{ \frac{\ZLp{n} (T')}{\ZLp{n}(T')} , \lambda \in \Delta_{d-1} \right\}
%\ees
%the set of $\lambda$s such that $(\lambda ,\pi')$ share with $T'$ the same first $n$ matrices in the Zorich cocycle.

%By unique ergodicity, 
%\bes
%diam(\Delta(\ZLp{n}(T'))) \xrightarrow{n \rightarrow \infty} 0, 
%\ees
%hence we can choose $k_1> k_0$ such that $diam(\Delta(\ZLp{n_{k_1}}(T')))\leq \log \nu /2$. Since clearly $\lambda' \in \Delta(\ZLp{n_{k_1}}(T'))$, by construction $\Delta(\ZLp{n_{k_1}}(T')\subset B_d (\underline{1}, \log \nu )$.
%Denote $B_{\overline{n}}\doteqdot \ZLp{n_{k_1}}(T') $, where $\overline{n} \doteqdot n_{k_1}$.

%Define
%\be \label{defdeltanu}
%\Delta_{\nu} \doteqdot \Delta \left( B_{\overline{n}} \, B_{\overline{n}} ) \right) .
%\ee
%%\be \label{defdeltanu}
%%\Delta_{\nu} \doteqdot \Delta \left( \ZLp{\overline{n}}(T') \, \ZLs(\gamma) \, \ZLp{\overline{n}}(T') ) \right) .
%\ee
%Remark that if $T\in \Delta_{\pi'} \times \{\pi'\}$, in particular the matrix $B_{\overline{n}}$ occurs twice the Zorich expansion of $T$.

%\paragraph{Definition of the generic set of IETs.}
%By Corollary \ref{corintDelta'}, applyed to $\Delta'=\Delta_{\nu}$, there exists a subset $\Delta_{\nu}^G=\Delta_{\nu}$ such that $\mu_{\Z}(\Delta_{\nu}^G \times \{\pi'\})=\mu_{\Z}(\Delta_{\nu} \times \{\pi'\} )>0$, such that for each $T \in \Delta_{\nu}^G \times \{\pi'\} $, 
%\be \label{logconseq}
%\lim_{l\rightarrow \infty} \frac{ \log \| \ZLs_{\Delta_{\nu}}^{(l)}{(T)} \|}{l} =0 . 
%\ee
%Let 
%%$G_{\nu} \doteqdot \nigcup_{k\in \mathbb{N}} \bigcup_{n\geq k} \Z^{k-n} (\Delta_{\nu}^G \times \{\pi'\})$
%$%G_{\nu} \doteqdot \bigcup_{n\in \mathbb{Z}} \Z^{-n} (\Delta_{\nu}^G \times \{\pi'\})$
%. Since $G_{\nu}$ is a 
%\Z -
%invariant set with positive measure, by ergodicity $\mu_{\Z}(G_{\nu})=1$.

%Repeat the previous construction for each $\nu_n = 1+ 1/n$ and define $G\doteqdot  \bigcap_{n\in \mathbb{Z}} G_{\nu_n}$. If $T\in G$, for each $\nu>1$ there exists $\widetilde{n}_0$ such that $\Z^{\widetilde{n}_0}T\in G_{\nu} \times \{ \pi' \}$. Let $\{ {\widetilde{n}_l} \}_{l \in \mathbb{N} } $ be the sequence of returns of $\Z^{\widetilde{n}_0}T$ to $G_{\nu} \times \{ \pi' \}$.
%The sequence of $\nu$-balanced induction times for $T$ in Proposition \ref{existencebalancedtimes} is given by the translated subsequence 
%$\{n_l | n_l \doteqdot {\widetilde{n}_l} + \overline{n} \}_{l \in \mathbb{N} } $.
%\be \label{nldef}
%\{n_l | n_l \doteqdot {\widetilde{n}_{2\overline{n}l + }} + \overline{n} \}_{l \in \mathbb{N} }.
%\ee
%\be \label{nldef}
%\{n_l | n_l \doteqdot {\widetilde{n}_l} + \overline{n} \}_{l \in \mathbb{N} }.
%\ee
%In the following paragraphs we verify that it satisfies properties (\ref{lengthsbalance}, \ref{heightsbalance}, \ref{positive}, \ref{controlnusB}, \ref{integrability}). 

%\paragraph{Integrability.}
%Remark that $\ZLs_{\Delta_{\nu}}^{(l)}{(Z^{\widetilde{n}_0} T)}= \ZLp{\widetilde{n}_l, \widetilde{n}_{l+1}}(T)$. Also, by (\ref{nldef}), 
%\bes
%\ZLp{\widetilde{n}_l, \widetilde{n}_{l+1}}(T) = (\ZLp{\overline{n}})^{-1} 
%\ZLp{\widetilde{n}_{2 \overline{n} l}, \widetilde{n}_{2\overline{n}l+ 2\overline{n}}}(T)
%\ZLp{\widetilde{n}_l, \widetilde{n}_{l+1}}(T) 
%\ZLp{\overline{n}}.
%\ees 
%Hence 
%\bes
%\frac{ \log \| \ZLp{\widetilde{n}_l, \widetilde{n}_{l+1}}(T)  \|}{l} \leq  2 \frac{ \log \max \{ \| \ZLp{\overline{n}}  \| , \| (\ZLp{\overline{n}})^{-1}  \| \} }{l} +  \sum_{j=0}^{2\overline{n}-1} \frac{\log \| \ZLs_{\Delta_{\nu}}^{(2\overline{n}l+j,2\overline{n}l+j +1 )}{(Z^{\widetilde{n}_0} T)} \|}{2\overline{n}l}2\overline{n} 
% \frac{ \log \| \ZLp{\widetilde{n}_l, \widetilde{n}_{l+1}}(T)  \|}{l} \leq  \frac{ \log \| (\ZLp{\overline{n}})^{-1}  \|}{l} +  \frac{\log \| \ZLs_{\Delta_{\nu}}^{(l)}{(Z^{\widetilde{n}_0} T)} \|}{l} +   \frac{\log \| \ZLp{\overline{n}}  \|}{l} .
%\ees
%Since $Z^{\widetilde{n}_0} T$ belongs $\Delta_{\nu}^G \times \{\pi'\}$, by (\ref{logconseq}) the RHS converges to zero as $l$ tends to infinity. 
%\paragraph{Positivity and balance of lengths and heights.}
%Let us prove that by our definition of $\{ n_l \}$
%\be\label{formB}
% \ZLp{{n}_l, {n}_{l+1}} =  \ZLp{\overline{n}}  B   \ZLp{\overline{n}} ,
%\ee
%for some $B\geq 0$. This implies immmediatly positivity (\ref{positive}).
%First remark that $n_{l+1}-n_{l} = \widetilde{n}_{2 \overline{n} l + 2\overline{n}} -\widetilde{n}_{2 \overline{n} l } = \geq 2 \overline{n}$ by definition of subsequence. By construction $\Z ^{\widetilde{n}_{2\overline{n}l }}T \in \Delta(B_{\overline{n}} B_{\overline{n}})$, hence $\Z ^{n} T  \in \Delta(B_{\overline{n}})$. Similarly  $\Z ^{\widetilde{n}_{2\overline{n}(l+1) - \overline{n} }}T \in \Delta(B_{\overline{n}})$. Since as remarked $\widetilde{n} _{2 \overline{n}} (l+1) -\overline{n} \geq  \widetilde{n} _{2 \overline{n} l } + \overline{n}$, this shows (\ref{formB}). 

%From (\ref{formB}) and (\ref{colprop}) it follows also that 
%\bes
%\nu_{row}(\ZLp{{n}_l, {n}_{l+1}} ) \leq \nu_{row}(B_{\overline{n}}); \qquad \nu_{col}(\ZLp{{n}_l, {n}_{l+1}} ) \leq \nu_{col}(B_{\overline{n}}).
%\ees
%Let us show that $\nu_{row}(B_{\overline{n}}) \leq \nu$ and $\nu_{col}(B_{\overline{n}}) \leq \kappa$ to finish the proof of (\ref{controlnusB}).

%Recall that by construction $B_{\overline{n}} (\Delta_{d-1})\subset B_d(\underline{1},\log \nu)$, hence for each $\underline{\lambda} \in \Delta_{d-1}$, by definition of $d_H (\overline{1}, \underline{\lambda})$, 
%\bes
%\frac{\max_{i=1,\dots , d}{\sum_j (B_{\overline{n}})_{ij} \lambda_j}}{\min_{i=1 , \dots ,d}{\sum_j (B_{\overline{n}})_{ij} \lambda_i}} \leq \nu .
%\ees
%Taking a sequence $\underline{\lambda}^n$ such that $\underline{\lambda}^n_i \xrightarrow{x\rightarrow \infty} \delta_{ik}$, we get  $\nu_{row}(B_{\overline{n}}) \leq \nu$. 
%Since $B_{\overline{n}} >0$, $\overline{B_{\overline{n}}^T (\Delta_{d-1})} \subset \Delta_{d-1}$, hence there exists $\kappa\geq 1$ such that $B_{\overline{n}}^T (\Delta_{d-1}) \subset B_d(\underline{1},\log \kappa)$. Reasoning as above, we get $\nu_{row}(B_{\overline{n}}^T) = \nu_{col}(B_{\overline{n}}) \leq \kappa$.

\subsubsection{Some consequences of balance.}
%\paragraph{Simple consequences of balance.}
We will also frequently use the following simple lemmas. Recall that $A>0$ means that $A$ has strictly positive entries.
\begin{lemma}\label{productpositive}
Let $A_i >0$, $A_i \in SL(d, \mathbb{Z})$ for $i=0, \dots, n$. If 
%$\underline{\lambda}'\in \Delta_{d}$ and 
$\underline{\lambda} = A_0 \cdot \underline{\lambda}'$, then $\sum_j\lambda_j > d \sum_j \lambda_j'$. 

If $\underline{h} = A_0 \cdot \underline{h}'$, then $\min_j h_j > d \min_j h_j'$. In particular if $\underline{h} = A_1 \cdots A_n \, \underline{e}$, then $\min_j {h_j} \geq d^n $.
\end{lemma}
\begin{proof}
All the properties follow easily from $A_{ij} \geq 1$.
%Just recall that $\lambda = \sum_{i} \underline{\lambda}_i$ and use that indeed $A_{ij} \geq 1$
%\bes
%\lambda =  \sum_{ij}  A_{ij} \, \underline{\lambda}'_j \geq 
%\sum_{i} \left( \sum_{j}  \underline{\lambda}'_j \right) 
% \geq 
%d \sum_{j}  \underline{\lambda}'_j =
% d \, \lambda'. 
%\ees
%The other is trivial.
\end{proof}

Recall that we defined $ \lambda^{(n_l)} \doteqdot \sum_j  \lambda^{(n_l)}_j $ and $ h^{(n_l)} \doteqdot \max_j  h^{(n_l)}_j$.

\begin{cor} \label{superexp} For each $L\in \mathbb{N}$, $\log(h^{(n_{L \overline{l}})}) \geq {L} \log d$.
In particular,
\be \label{superexplimit}
\lim_{l\rightarrow +\infty} \frac{1}{\log h^{(n_{l})} }=0.
%\log(h^{(n_{L \overline{l}})}) \geq {L} \log d
\ee
%In particular, $\lim_{l\rightarrow +\infty} \frac{1}{\log h^{(n_{l})} }=0$.
\end{cor}

\begin{proof}
For the first property, apply Lemma \ref{productpositive} to $h^{(n_{L \overline{l}})}= {\ZLp{n_{L \overline{l}}}}^T \underline{e}$ and remark that ${\ZLp{n_{L \overline{l}}}}^T $ is the product of at least $L$ positive matrices by Property 3 (see (\ref{positive})) of the sequence $\{n_l\}_{l\in \mathbb{N}}$. It follows that $\log h^{(n_{l})} \geq \log d  [{l}/{\overline{l}}]  %\left[\frac{l}{\overline{l}}\right] 
$
 and hence we get (\ref{superexplimit}).
\end{proof}


Recall that we defined $ \lambda^{(n_l)} \doteqdot \sum_j  \lambda^{(n_l)}_j $ and $ h^{(n_l)} \doteqdot \max_j  h^{(n_l)}_j$.
\begin{lemma}\label{balancehlambda}
If $n_l$ has $\nu$-balanced lengths (\ref{lengthsbalance}) and $\kappa$-balanced heights (\ref{heightsbalance}), then for each $j=0,\dots, d$,
%%%%ELIMINABILE: due in una
\begin{eqnarray}
\frac{1}{\kappa \lambda^{(n_l)} } &\leq &h^{(n_l)}_j \leq \frac{\kappa}{ \lambda^{(n_l)} } ;\label{hcomparedtolambda} \\ 
\frac{1}{d \kappa \nu h^{(n_l)} } &\leq & \lambda^{(n_l)}_j \leq \frac{\kappa}{h^{(n_l)} } .\label{lambdacomparedtoh} 
\end{eqnarray}
\end{lemma}
Hence, if $n_l$ is a balanced time, $\lambda^{(n_l)} \approx 1/ h^{(n_l)}$ up to constants.
%%%ELIMINABILE
\begin{proof}
Since by Kac lemma $\sum_i h^{(n_l)}_i \lambda^{(n_l)}_i =1$, remark that $\min_i h^{(n_l)}_i \lambda^{(n_l)} \leq 1 $ and $\max_i h^{(n_l)}_i \lambda^{(n_l)} \geq 1 $. Together with heights balance (\ref{heightsbalance}), this gives
\bes
\frac{1}{\kappa \lambda^{(n_l)} } \leq \frac{1}{\kappa}  \max_i h^{(n_l)}_i  \leq h^{(n_l)}_j \leq {\kappa} \min_i h^{(n_l)}_i \leq \frac{\kappa}{ \lambda^{(n_l)} }.
\ees 
To show (\ref{lambdacomparedtoh}), let $i$ be such that $ h^{(n_l)}_i \lambda^{(n_l)}_i = \max_j h^{(n_l)}_j \lambda^{(n_l)}_j \geq  1/d$. Also, for each $j$, $h^{(n_l)}_j \lambda^{(n_l)}_j <1$. Using also lengths balance (\ref{lengthsbalance}),
\bes
\frac{1}{\kappa \nu d h^{(n_l)}_{\phantom{j}} } \leq  \frac{1}{\nu d h^{(n_l)}_i }  \leq  \frac{\lambda^{(n_l)}_i}{\nu} \leq \lambda^{(n_l)}_j \leq  \frac{1 }{h^{(n_l)}_j} \leq \frac{\kappa}{ h^{(n_l)}_{\phantom{j}} }.
\ees 
\end{proof}
\begin{lemma} \label{loglimit}
For each fixed $L\in \mathbb{N}$, 
\be
\lim_{l\rightarrow + \infty } \frac{ \log \|\ZLp{n_l, n_{l+L}} \|}{\log h^{(n_l)}} = \lim_{l\rightarrow + \infty } \frac{ \log \|\ZLp{n_{l-L}, n_l} \|}{\log h^{(n_l)}} = 0.
\ee
\end{lemma}
\begin{proof}
By Corollary \ref{superexp}, $\log h^{(n_{l })} \geq {[l/\overline{l}]} \log d$, where $[\cdot]$ denotes the integer part.
Hence
\be
\lim_{l\rightarrow + \infty } \frac{ \log \|\ZLp{n_l, n_{l+L}} \|}{\log h^{(n_l)} } \leq  \lim_{l\rightarrow + \infty }
%\leq \|\ZLp{n_{l_{-2}},n_{l_0} } \|  \lambda^{(n_{l_0})} \leq \prod_{l=l_{-2}}^{l_0-1} \left\| \ZLp{n_{l},n_{l+1} } \right\|  \lambda^{(n_{l_0})
\frac{\sum_{i=l}^{l + L-1} \log \left\| \ZLp{n_{i},n_{i+1} } \right\|} {(l/\overline{l} -1 )\log d  } .
%\frac{l+L}{l/l_0 -1} \log d  .
\ee
%Recall that $ l_{0}=l_{-2}+L $ (see (\ref{nl1nl2def})), hence the terms in the sum in the LHS of (\ref{loglambdas}) are at most $L$.
Using the property (\ref{integrability}), each of the $L$ terms in the sum in the RHS can be bounded for $l\gg1 $ by $\tau \log i \leq \tau \log (l+L-1)$ and hence the first limit is zero. The second one is analogous.
\end{proof}

\subsection{Growth of Birkhoff sums of $u$.}\label{growthBSusec}
%of the first derivative.}\label{growthBS1sec}
Assume in this section that the roof function is $u(x)=1/x$. 
%The roof function $v(x)$ will be treated similary and results for the original $f(x)$ will be deduced in the next section.
\subsubsection{Growth of Birkhoff sums along a balanced tower.}\label{BSalongtowersec}
In order to understand the asymptotic growth of $\BS{u}{r}(x)$, we first consider $\BS{u}{r}(x)$ when $x \in I^{(n)}_j$ is a point in the base of the tower $Z^{(n)}_j$, $r=h^{(n)}_j$ is exactly the height of the same tower and $n$ is one of the balanced return times constructed in Section \ref{existencebalancedtimes}. This preliminary estimate is used in Section \ref{generalx} as a building block to get an estimate for any $r$ and most of the other points $x$.    

%Assume that the roof function is $u(x)=C/x$. 
\begin{prop}\label{alongtower}
Assume $T\in \M^+$. Let $n_{l_0}$ be a balanced return time given by Proposition \ref{existencebalancedtimes}. Let $x_0 \in I^{(n_{l_0})}_{j_0}$ be a point belonging to the base of the tower $Z^{(n_{l_0})}_{j_0}$ and let $r_0= h^{(n_{l_0})}_{j_0}$ be the corresponding tower height. Given $\varepsilon>0$, there exists $l(\varepsilon)$ such that for $l_0 \geq l(\varepsilon)$,
\be \label{alongtowerestimate}
(1-\varepsilon)  h^{(n_{l_0})}_{j_0} \ln 
\left( 
h^{(n_{l_0})} 
\right) 
\leq \BS{u}{{r_0}}(x_0) - \frac{1}{x_0}
\leq  (1+ \varepsilon) h^{(n_{l_0})}_{j_0} \ln \left( h^{(n_{l_0})} \right).
\ee
\end{prop}

A Birkhoff sum of the form $\BS{u}{{r_0}}(x_0)$ where $x_0$ and $r_0$ satisfy the hypotheses of Proposition \ref{alongtower} will be referred as \emph{Birkhoff sum along a tower}. The proposition shows that each sum along a tower gives a contribution of order $r_0\log(r_0)$, plus the contribution of the closest point to the singularity, $x_0$, which could be arbitrary large and will be estimated separately when using this sums as building blocks in Section \ref{generalx}.

%%%ELIMINABILE
%\paragraph{An idea of the proof of Proposition \ref{alongtower}.}
%We want to 
%compare $(\BS{u}{{r_0}}(x_0) -1/x_0)/ h^{(n_{l_0})} $ with the integral of the roof function $u(x)=1/x$. 
%We will show that one can find $L$ independent of ${l_0}$ so that if we consider the previous balanced induction time $n_{l_0-L}$, the main contribution to the Birkhoff sum $\BS{u}{{r_0}}(x_0)$ is determined by the visits of $\{ T^{r}(x_0) \} _{0 \leq r < r_0}$ to $[\lambda^{(n_{l_0-L})},1]$
%. We first find $L'\leq L$ such that points of the orbit $\{ T^r(x_0)\} _{0\leq r \leq r_0}$ are uniformly distributed inside each element of the partition $\xi_{n_{l_0-L'}}$ into past towers floors.
%This allows
%to compare the main contribution to the Birkhoff sum to a Riemannian sum which is close to the integral $ \int_{[\lambda^{(n_{l_0-L})},1]}$
%$\frac{1}{x} d x = \log (1 / \lambda^{(n_0-L)})$ and then compare it to $\log (h^{(n_0)}) $. The paragraph about the Gap Error at page \pageref{estimategapsec} is devoted to show that the reminder, given by visits of $\{ T^r(x_0)\} _{0\leq r < r_0}$ to  $I^{(n_{l_0-L})} $,
% is negligible compared to the main contribution.

\begin{proofof}{Proposition}{alongtower}
%%%ELIMINABILE
Consider the inducing intervals $I^{(n_{l})} = [0,\lambda^{(n_{l})}]$ 
%%%ELIMINABILE
%corresponding to the subsequence of balanced induction times 
where $\{n_l\}_{l\in \mathbb{N}}$ is the sequence of balanced induction times constructed in Section \ref{existencebalancedtimessec}.
%%%
%Since they are nested, we can define the intervals $\Lambda^{(n_l)} = I^{(n_{l})}\backslash I^{(n_{l+1})} = (\lambda^{(n_{l+1})},  \lambda ^{(n_{l})} ]$. For each $n_{l_{-2}}$, the original interval is then partitioned as
%\bes
%I=  I^{(n_{l_0})} \cup ( I^{(n_{l_{-2}})} \backslash  I^{(n_{l_0})}  \cup \up_{l=0}^{l_{-2}} \Lambda^{(n_l)} \cup (\lambda^{(l_0)},1].
%\ees
%Remark that $diam(\widetilde {\ZLp{n_{\overline{l}}}} \Delta_{d-1}) < \infty$ by (\ref{positive}).
Given any $\epsilon > 0$, let $D$ and $\overline{l}$ be given by Proposition \ref{existencebalancedtimes}. Choose $L_1 \in \mathbb{N}$ such that 
%%%ELIMINABILE con ref
\be \label{L_1def}
(1-e^{-D})^{L_1-1} D  %diam_H \left( \widetilde {\ZLp{ n_{\overline{l}}}}\Delta_{d-1} \right)
 <\epsilon .
\ee
%where $D$ and $\overline{l}$ are given by Proposition \ref{existencebalancedtimes} and w
We remark that $diam(\widetilde {\ZLp{n_{\overline{l}}}} \Delta_{d-1}) < \infty$ by (\ref{positive}). Choose also $ L_2 \in \mathbb{N}$ such that $1/d^{L_2} <\epsilon$. Assume $l_0 \geq \overline{l}(1 + L_1 + L_2)$. For convenience, introduce the notation:
%(where $\overline{l}$ is also given by Proposition \ref{existencebalancedtimes})
\be \label{nl1nl2def}
{l_{-1}} \doteqdot {l_0 -L_1 \overline{l}};\qquad {l_{-2}} \doteqdot {l_{-1} -L_{2}\overline{l}} = {l_0  - ( L_1 + L_2) \overline{l} } ; 
%\quad \mathrm{so} \quad n_{l_{-1}} \doteqdot n_{l_0 -L_1 \overline{l}};\qquad n_{l_{-2}} \doteqdot n_{l_{-1} -L_2\overline{l}}.
\ee
The past induction times $n_{l_{-1}}$ and $n_{l_{-2}}$ will play the following role in the proof: $n_{l_{-1}}$ is such that the elements of the orbit $\{ T^r(x_0) \} _{0 \leq r < r_0}$ are uniformly distributed inside the elements of the partition $\xi_{n_{l_{-1}}}$; $n_{l_{-2}}$ is such that the main contribution to $\BS{u}{{r_0}}(x_0)$ comes from visits to $(\lambda^{(n_{l_{-2}})},1]$.

Denote the points along the $T$-orbit of $x_0$ by $x_i= T^i (x_0) $, $0\leq i < r_0$. 
Since the original interval can be partitioned as
%%%ELIMINABILE
\bes
I^{(0)} =  I^{(n_{l_0})} \cup  I^{(n_{l_{-2}})} \backslash  I^{(n_{l_0})} \cup (\lambda^{(n_{l_{-2}})},1],
\ees
and $x_i \notin I^{(n_{l_0})} $, for $1 \leq i < r_0$ because $r_0$ is by definition the first return time of $x_0$ to $I^{(n_{l_0})}$, the Birkhoff sums can be decomposed as follows:
\be \label{BSdecomp}
\BS{u}{{r_0}}(x_0) = \sum_{i=0}^{r_0 -1 } \frac{1}{x_i} =  \frac{1}{x_0} + \sum_{x_i \in I^{(n_{l_{-2}})} \backslash  I^{(n_{l_0})}} \frac{1}{x_i} +\sum_{x_i \in (\lambda^{(n_{l_{-2}})},1]} \frac{1}{x_i} .
\ee
We will refer to the first term in the RHS of (\ref{BSdecomp}) as \emph{singular error}, to the sum which appears as a second term as \emph{gap error},  while the sum which appears as a last term determines the \emph{main contribution}.
%, to the first term as \emph{singular error}, while the sum which appears as a second term in the LHS will be called \emph{gap error}.

%\paragraph{Estimate of the Singular Error.}
%We will first show that the singular error in (\ref{BSdecomp}) is negligible compared to the main order in Proposition \ref{alongtower}, since we will need some of the techniques used in this Section also for the estimate of the main contribution (in Section \ref{estimatemainsection}). More precisely, let us prove the following. 
%\begin{lemma}[Singular error.]\label{estimategaplemma}
%For each $\epsilon$, if $r_0 > R_0(\epsilon)$,
%\be \label{estimategap}
% \left| \sum_{x_i \in I^{(n_{l_{-2}})} \backslash  I^{(n_{l_0})}}   \frac{1}{x_i}\, \right| \leq
% \epsilon    \frac{1}{ \lambda^{(n_{l_0})} }  \log \frac{1}{\lambda^{(n_{l_0})}} .
%\ee 
%\end{lemma}
 
%%%ELIMINABILE
\paragraph{Uniform ergodic convergence.}
The following Lemma gives an uniform bound for the equidistribution of the orbits of points $x\in I^{(n_{l_0})}$ inside the elements of the past partition $\xi_{n_{l_{-1}}}$. %%%The proof is based on arguments similar to the ergodic theorem for Markov chains.
Recall that $\ZLp{ n_{l_{-1}},n_{l_0} }_{i j}$ gives the number of visits of $x \in I^{(n_{l_0})}_{j} $ to $I^{(n_{l_{-1}})}_{i}$ before the time $h^{(n_{l_0})}_j$ of first return to $I^{(n_{l_0})}$.
%%%
\begin{lemma}[Uniform distribution] \label{uniformdistributionlemma}
For each $1\leq i,\, j\leq d $,
\be \label{uniformdistribution}
e^{-2\epsilon} \lambda^{(n_{l_{-1}})}_i  \leq \frac{\ZLp{ n_{l_{-1}},n_{l_0} }_{i j}}{h^{(n_{l_0})}_j} \leq e^{2 \epsilon} \lambda^{(n_{l_{-1}})}_i .
\ee
\end{lemma}
\begin{proof}
Consider the sets $\widetilde {\ZLp{ n_{l_{-1}},n } } \Delta_{d-1} \subset \Delta_{d-1}$, for $n > n_{l_{-1}}$, which form a nested sequence of compact sets. 
%(if $l_{-1} \geq \overline{l}$)
By the transformation formula (\ref{lengthsrelationR}) for lengths vectors $\underline{\lambda}^{(n_{l_{-1}})} = \ZLp{ n_{l_{-1}},n }  \underline{\lambda}^{(n)}$, the renormalized vector
\bes
\frac{ \underline{\lambda}^{(n_{l_{-1}})}} { \lambda^{(n_{l_{-1}})} } \in \bigcap_{n > n_{l_{-1}}} \widetilde {\ZLp{ n_{l_{-1}},n } } \Delta_{d-1} .
\ees
When $n= n_{l_0}$, since $l_0 = l_{-1} + L_1 \overline{l}$, applying $L_1$ times Property 3 (positivity) in Proposition \ref{existencebalancedtimes} through the contraction property (\ref{contraction}), we get
% $ \ZLp{ n_{l_{-1}},n_{l_0} }$ is a product of $L_2$ positive matrices with definite contraction properties,
\be \label{contracteddiamater}
diam_H ( \widetilde {\ZLp{ n_{l_{-1}},n_{l_0} } } \Delta_{d-1} ) \leq (1-e^{-D})^{L_1-1} D 
%diam_H(\widetilde {\ZLp{ n_{l_{-1}}} } \Delta_{d-1}) 
\leq \epsilon , 
\ee
where the last inequality %%%ref
follows by the choice (\ref{L_1def}) of $L_1$.%, since $l_{-1} \geq \overline{l}$.

Denote by $\underline{e}_j$ the unit vector $(\underline{e}_j)_i=\delta_{ij}$ ($\delta$ is here the Kronecker symbol). Since both the vectors $\widetilde { \ZLp{ n_{l_{-1}},n_{l_0} }} \underline{e}_j $ and $ \frac{ \underline{\lambda}^{(n_{l_{-1}})}} { \lambda^{(n_{l_{-1}})} } $ belong to the closure of ${\ZLp{ n_{l_{-1}},n_{l_0} } } { \Delta_{d-1} }$,
%\bes
%\widetilde { \ZLp{ n_{l_{-1}},n_{l_0} }} \underline{e}_j = \frac{\ZLp{ n_{l_{-1}},n_{l_0} }_{ij} }{ \sum_{k}\ZLp{ n_{l_{-1}},n_{l_0} }_{kj}}, \quad  \frac{ \underline{\lambda}^{(n_{l_{-1}})}} { \lambda^{(n_{l_{-1}})} }  \in  {\ZLp{ n_{l_{-1}},n_{l_0} } } \overline{ \Delta_{d-1} },
%\ees
it follows by (\ref{contracteddiamater}), using compactness, that% and continuity of the distance, that
\bes
d_H \left(  \frac{ \underline{\lambda}^{(n_{l_{-1}})}} {\lambda^{(n_{l_{-1}})} }  , 
\widetilde { \ZLp{ n_{l_{-1}},n_{l_0} } } \underline{e_j}  \right) = \log \frac{\max_{i=1,\dots , d}\frac{\ZLp{ n_{l_{-1}},n_{l_0} }_{ij}}{{\lambda_i^{(n_{l_{-1}})}} }} {\min_{i=1 , \dots ,d}\frac{\ZLp{ n_{l_{-1}},n_{l_0} }_{ij}}{{\lambda_i^{(n_{l_{-1}})}} }}    \leq \epsilon , 
\ees
where we also used the invariance of the distance expression by multiplication of the arguments by a scalar.
Equivalently, for each $1\leq i ,\, k \leq d$, 
\be
\label{projdistancecons}
e^{-\epsilon} \left( {\ZLp{ n_{l_{-1}},n_{l_0} }_{kj}}{{\lambda_i^{(n_{l_{-1}})}} }\right)
 \leq {\ZLp{ n_{l_{-1}},n_{l_0} }_{ij}}{{\lambda^{(n_{l_{-1}})}}_k } \leq e^{\epsilon}\left( {\ZLp{ n_{l_{-1}},n_{l_0} }_{kj}}{{\lambda_i^{(n_{l_{-1}})}} } \right)
\ee
and summing over $k$ we get
\be\label{1stratio}
e^{-\epsilon}\leq  \frac{{\ZLp{ n_{l_{-1}},n_{l_0} }_{ij}}{{\lambda^{(n_{l_{-1}})}} } }{{\sum_k \ZLp{ n_{l_{-1}},n_{l_0} }_{kj}}{{\lambda_i^{(n_{l_{-1}})}} } }  \leq e^{\epsilon} .
%\left| \frac{{\ZLp{ n_{l_{-1}},n_{l_0} }_{ij}}{{\lambda^{(n_{l_{-1}})}} } }{{\sum_k \ZLp{ n_{l_{-1}},n_{l_0} }_{kj}}{{\lambda_i^{(n_{l_{-1}})}} } } -1 \right| \leq e^{\epsilon} .
\ee
If we multiply (\ref{projdistancecons})
 by $h^{(n_{l_{-1}})}_i$ and then sum also over $i$, using that $\sum_i h^{(n_{l_{-1}})}_i \lambda ^{(n_{l_{-1}})}_i=1$ by Kac's lemma and that $\underline{h}^{(n_{l_0})}= {\ZLp{ n_{l_{-1}},n_{l_0} }}^T \underline{h}^{(n_{l_{-1}})}$,
\be\label{2ndratio}
e^{-\epsilon} \leq  \frac{ {h}^{(n_{l_0})}_j {\lambda^{(n_{l_{-1}})}}}{ { \sum_k \ZLp{ n_{l_{-1}},n_{l_0} }_{kj}}}  \leq e^{\epsilon} .
%\left| \frac{ {h}^{(n_{l_0})}_j {\lambda^{(n_{l_{-1}})}}}{ { \sum_k \ZLp{ n_{l_{-1}},n_{l_0} }_{kj}}} -1 \right| \leq e^{\epsilon} .
\ee
The combination of (\ref{1stratio}, \ref{2ndratio}) 
%and a suitable choice of $\epsilon$
% such that $e^{2\epsilon} \leq 1+\varepsilon$
 gives (\ref{uniformdistribution}).
\end{proof}


\paragraph{Estimate of the Main Contribution.}\label{estimatemainsec}
The following lemma shows that the main contribution in (\ref{BSdecomp}) determines the order of the Birkhoff sum in Proposition \ref{alongtower}.
\begin{lemma}[Main contribution.]\label{estimatemainlemma}
For each $\epsilon>0$, if $l_0 > l_{m}(\epsilon)$,
\be \label{estimatemain}
e^{-2\epsilon} {(1-\epsilon)^2 }  {h_{j_0}^{(n_{l_0})} }  \log h^{(n_{l_0})}  \leq 
%\sum_{F_{\alpha} \subset  (\lambda^{(n_{l_{-2}})},1] } \sum_{x_i \in F_{\alpha}}  \frac{1}{x_i}
\sum_{x_i \in (\lambda^{(n_{l_{-2}})},1]} \frac{1}{x_i} 
% \sum_{x_i \in I^{(n_{l_{-2}})} \backslash  I^{(n_{l_0})}} \frac{1}{x_i}
\leq  
e^{2\epsilon} (1+ \epsilon)^2    {h_{j_0}^{(n_{l_0})} }  \log h^{(n_{l_0})}
% \frac{1}{ \lambda^{(n_{l_0})} }  \log \frac{1}{\lambda^{(n_{l_0})}} 
\ee 
\end{lemma}
\begin{proof}
Consider the partition $\xi_{n_{l_{-1}}}\left( (\lambda^{(n_{l_{-2}})},1]\right)$ of $(\lambda^{(n_{l_{-2}})},1]$. Recall that the elements $F_{\alpha}\in \xi_{n_{l_{-1}}}$ are floors of the towers of step $n_{l_{-1}}$, i.e. for some $1\leq j_{\alpha} \leq d$, $F_{\alpha}= T^k( I^{(n_{l_{-1}})}_{j_{\alpha}})$, where $0\leq k< h^{(n_{l_{-1}})}_{j_{\alpha}}$. In particular $Leb(F_{\alpha})= \lambda^{(n_{l_{-1}})}_{j_{\alpha}}$. For each $F_{\alpha}$ choose, by the mean value theorem, a point $\bar{x}_{\alpha}$ such that
\be\label{xalphadef}
\frac{1}{\bar{x}_{\alpha}} \doteqdot \frac{1}{ \lambda^{(n_{l_{-1}})}_{j_{\alpha}}} \int_{F_{\alpha}} \frac{1}{s} \ud{s}.
\ee

\begin{lemma}\label{ratios}
If $x_i \in F_{\alpha}$ and $F_{\alpha} \subset (\lambda^{(n_{l_{-2}})},1]$,
\bes
1-\epsilon \leq \frac{1/x_i}{1/ \bar{x}_{\alpha}}\leq 1+\epsilon.
\ees
\end{lemma}
\begin{proof}
Let $F_{\alpha}=[a,b)$; then $a\leq x_i, \, \bar{x}_{\alpha} <b$. Since by assumption $b-a \leq \lambda^{(n_{l_{-1}})}$ and $a\geq \lambda^{(n_{l_{-2}})} $, 
\begin{eqnarray}
%%%ELIMINABILE in due righe
\frac{\bar{x}_{\alpha}}{ x_i} &\leq& \frac{b}{a}= \frac{a+(b-a)}{a} \leq 1 + \frac{\lambda^{(n_{l_{-1}})}}{\lambda^{(n_{l_{-2}})}};
\nonumber \\
\frac{\bar{x}_{\alpha}}{ x_i} &\geq& \frac{a}{b} = \frac{b-(b-a)}{b} \geq 1 - \frac{\lambda^{(n_{l_{-1}})}}{\lambda^{(n_{l_{-2}})}} .\nonumber
\end{eqnarray}
Let us show that $\lambda^{(n_{l_{-1}})}/ \lambda^{(n_{l_{-2}})}< \epsilon$. Since $l_{-1}=l_{-2}+{L_2}\overline{l}$,
\bes
\underline{\lambda}^{(n_{l_{-2}})} = \prod_{i=0}^{L_2-1} \ZLp{ n_{l_{-2} +i \overline{l}}, n_{l_{-2} + (i+1) \overline{l} }} \underline{\lambda}^{(n_{l_{-1}})}
%\ZLp{ n_{l_{-2}}, n_{l_{-2}} } \underline{\lambda}^{(n_{l_{-1}})} =  \ZLp{ n_{l_{-2}}, n_{l_{-2} + 1 } } \cdot \dots \cdot  \ZLp{ n_{l_{-1}-1}, n_{l_{-1}} } \underline{\lambda}^{(n_{l_{-1}})}
\ees
and each of the matrices in the product has 
%the matrices $\ZLp{ n_{l}, n_{l+1} }$ are 
positive entries by Property 3 in Proposition \ref{existencebalancedtimes}. Hence by iterated application of Lemma \ref{productpositive} and by the choice of $L_2$, we get $\lambda^{(n_{l_{-1}})}/ \lambda^{(n_{l_{-2}})}< 1/d^{L_2} < \epsilon$.  
\end{proof}
Rearranging the main contribution in (\ref{BSdecomp}) by floors, i.e.
\bes
\sum_{x_i \in (\lambda^{(n_{l_{-2}})},1]}  \frac{1}{x_i}  = \sum_{F_{\alpha} \in  \xi_{n_{l_{-1}}} \left( (\lambda^{(n_{l_{-2}})},1]\right) } \sum_{x_i \in F_{\alpha}}  \frac{1}{x_i},
\ees
and applying Lemma \ref{ratios}, we get
\be \label{mainbetweenFalphas}
%(1-\epsilon) 
 \sum_{F_{\alpha} \subset  (\lambda^{(n_{l_{-2}})},1] } \sum_{x_i \in F_{\alpha}}  \frac{1-\epsilon}{\bar{x}_{\alpha}}  \leq \sum_{F_{\alpha} \subset  (\lambda^{(n_{l_{-2}})},1] } \sum_{x_i \in F_{\alpha}}  \frac{1}{x_i}\leq  %(1+\epsilon) 
 \sum_{F_{\alpha} \subset  (\lambda^{(n_{l_{-2}})},1] } \sum_{x_i \in F_{\alpha}}  \frac{1+\epsilon}{\bar{x}_{\alpha}} .
\ee
Consider now 
\be \label{termmain}
\sum_{x_i \in F_{\alpha}}  \frac{1}{\bar{x}_{\alpha}} = \#\{ x_i \in F_{\alpha} \}  \frac{1}{\bar{x}_{\alpha}}.
\ee
%where $\#$ denotes the cardinality of a set. 
Recall that $x_0\in I^{(n_{l_0})}_{j_0} $; if $F_{\alpha}= T^k( I^{(n_{l_{-1}})}_{j_{\alpha}})$ is a floor of the $j^{th}$ tower $Z^{(n_{l_{-1}})}_{j_{\alpha}}$, 
\be \label{cardinality}
\#\{ x_i \in F_{\alpha} \} =  \#\{ x_i \in I^{(n_{l_{-1}})}_{j_{\alpha}}  \} = \ZLp{ n_{l_{-1}},n_{l_0} }_{j_{\alpha} j_0}.
\ee
In (\ref{cardinality}) we used the dynamical meaning of $\ZLp{ n_{l_{-1}},n_{l_0} }_{j_{\alpha} j_0}$ 
%, which gives exactly the number of visits of $x_0\in I^{(n_{l_0})}_{j_0} $ to $I^{(n_{l_{-1}})}_{j_\alpha}$ before the first return to $I^{(n_{l_0})}$, 
together with the fact that $Z^{(n_{l_0})}_{j_0}$ is decomposed in a whole number of elements of $\xi_{n_{l_{-1}}}$ corresponding to towers of the previous step $n_{l_{-1}}$,  hence visits to a floor $T^k 
%\left( 
I^{(n_{l_{-1}})}_{j_{\alpha}} 
%\right)
$ of the tower are in one to one correspondence with visits to its base $I^{(n_{l_{-1}})}_{j_{\alpha}}$. 

%By construction of the sequence of balanced times (see Section \ref{existencebalancedtimessec}), 
%\bes
%e^{-2\epsilon} \leq \frac{\ZLp{ n_{l_{-1}},n_{l_0} }_{j_{\alpha} j}}{\ZLp{ n_{l_{-1}},n_{l_0} }_{j_{\alpha} j_0}} \leq \nu .
%\ees

%Applying this estimate to the $j_{\alpha}$-component of $\underline{\lambda}^{n_{l_{-1}}} = \ZLp{ n_{l_{-1}},n_{l_0} } \underline{\lambda}^{n_{l_0}} $ and using that $\lambda^{(n_{l_0})}= \sum_j \lambda^{(n_{l_0})}_j $, we get
%\bes 
%e^{-2\epsilon}  \ZLp{ n_{l_{-1}},n_{l_0} }_{j_{\alpha} j_0} \lambda^{(n_{l_0})} \leq \lambda^{(n_{l_{-1}})}_{j_{\alpha}}  =  \sum_j \ZLp{ n_{l_{-1}},n_{l_0} }_{j_{\alpha} j}  \lambda^{(n_{l_0})}_{j} \leq \nu   \ZLp{ n_{l_{-1}},n_{l_0} }_{j_{\alpha} j_0} \lambda^{(n_{l_0})}, 
%\ees
%which can be combined to (\ref{cardinality}) to get

From Lemma \ref{uniformdistributionlemma} and (\ref{cardinality}),
\be
e^{-2\epsilon} {\lambda^{(n_{l_{-1}})}_{j_{\alpha}} } h^{(n_{l_0})}_{j_0} \leq \#\{ x_i \in F_{\alpha} \} \leq {e^{2\epsilon}} {\lambda^{(n_{l_{-1}})}_{j_{\alpha}} } {h^{(n_{l_0})}_{j_0} }.
\ee
Using this bound and recalling the definition (\ref{xalphadef}) of $\overline{x}_{\alpha}$, we get% for the term (\ref{termmain}) the estimate
\bes
e^{-2\epsilon} { h^{(n_{l_0})}_{j_0} }  \int_{F_{\alpha}} \frac{1}{s} \ud  s  \leq \#\{ x_i \in F_{\alpha} \}  \frac{1}{\bar{x}_{\alpha}} \leq {e^{2\epsilon}} {h^{(n_{l_0})}_{j_0} }  \int_{F_{\alpha}} \frac{1}{s} \ud  s.
\ees
Summing it over $F_{\alpha} \subset  (\lambda^{(n_{l_{-2}})},1]$ and using
\bes
 \sum_{F_{\alpha} \subset  (\lambda^{(n_{l_{-2}})},1] }   \int_{F_{\alpha}} \frac{1}{s} \ud  s =  \int_{(\lambda^{(n_{l_{-2}})},1]} \frac{1}{s}\ud s = \log \frac{1}{\lambda^{(n_{l_{-2}})}} , 
\ees
we get by (\ref{mainbetweenFalphas})
\be \label{mainwlambda}
{e^{-2\epsilon}}(1-\epsilon)   h^{(n_{l_0})}_{j_0}   \log \frac{1}{\lambda^{(n_{l_{-2}})}}  \leq \sum_{F_{\alpha} \subset  (\lambda^{(n_{l_{-2}})},1] } \sum_{x_i \in F_{\alpha}}  \frac{1}{x_i}\leq  
 {e^{2\epsilon}}  (1+\epsilon)  h^{(n_{l_0})}_{j_0}   \log \frac{1}{\lambda^{(n_{l_{-2}})}} .
\ee

In order to get the estimate (\ref{estimatemain}) of Lemma \ref{estimatemainlemma} from (\ref{mainwlambda}) it is enough to compare $1/\lambda^{(n_{l_{-2}})}$ with $h^{(n_{l_0})}$. Since $\lambda^{(n_{l_{-2}})}> \lambda^{(n_{l_0})} \geq 1/(\kappa h^{(n_{l_0})} )$ by $\kappa$-balance of heights (see Lemma \ref{balancehlambda}), we get 
\bes
\log (1/ \lambda^{(n_{l_{-2}})}) \leq   \log(h^{(n_{l_0})}) \left( 1+ \log{\kappa}/{ \log(h^{(n_{l_0})}) } \right)
% \log (\frac{1}{ \lambda^{(n_{l_{-2}})}}) \leq   \log(h^{(n_{l_0})}) \left( 1+\frac{\log \kappa}{ \log(h^{(n_{l_0})}) } \right)
\ees
and, if $l_0\geq l_m$ for some $l_m(\epsilon)>0$, the upper estimate in (\ref{estimatemain}) by Corollary \ref{superexp}.
%grows at least exponentially (Lemma \ref{superexp}).
%\be \label{superexp}
%\log(h^{(n_{l_0})}) \geq \frac{l_0}{\overline{l}}\log d
%\ee
% by Lemma \ref{productpositive} and Property 3, i.e. (\ref{positive}), of the balanced times.

For the lower bound, adding and subtracting $h ^{(n_{l_0})}_{j_0} \log  h^{(n_{l_0})}$,
\be \label{lambdavsh}
 h^{(n_{l_0})}_{j_0}   \log  \frac{1}{\lambda^{(n_{l_{-2}})}}  =   h^{(n_{l_0})}_{j_0} \log h^{(n_{l_0})}  \left( 1  -   \frac{ \log \left( h^{(n_{l_0})}_{j_0}  \lambda^{(n_{l_{-2}})}\right)}{  \log h^{(n_{l_0})}} \right).
%\geq   { \log \frac{1}{\lambda^{(n_{l_0})}}} }{ h^{(n_{l_0})} } \right)
%\left(
% 1- \frac{ 
%\left|
%\log \left( \frac{ \lambda^{(n_{l_0})}}{\lambda^{(n_{l_{-2}})}} \right)  
%\right| 
%} {  \log \frac{1}{ \lambda^{(n_{l_0})}}  } 
%\right)
\ee
In order to estimate the very last term in (\ref{lambdavsh}), notice that again by Lemma \ref{balancehlambda} %\ref{productpositive} 
and balance, $\lambda^{(n_{l_{-2}})} \leq \kappa /  h^{(n_{l_{-2}})} $. Hence, using that $h^{(n_{l_0})}/  h^{(n_{l_{-2}})} \leq \| \ZLp{n_{l_{-2}},n_{l_0} } \|$,
\be \label{logsumisepsilon}
\frac{ \log \left( h^{(n_{l_0})}  \lambda^{(n_{l_{-2}})}\right)}{  \log h^{(n_{l_0})}}  \leq 
\frac{\log \kappa + \log  \| \ZLp{n_{l_{-2}}
%{l_0 - L_1 \overline{l}}
,n_{l_0} } \| }{  \log h^{(n_{l_0})}}.
%\overline{l}  \frac{ \log d \kappa }{ {l_0} \log d }  +\overline{l}  \frac{ \log \frac{h^{(n_{l_0})}}{ h^{(n_{l_{-1}})}}} { {l_0} \log d }
\ee
Enlarging  $ l_{m}$ if necessary, the RHS is less than $\epsilon$ for $l_0\geq l_m$  by Lemma \ref{loglimit} 
(recall that the difference $l_{0}-l_{-2}=(L_1 + L_2) \overline{l}$ is fixed) and Corollary \ref{superexp}.
%as long as $l \geq l_{m}$ for some $ l_{m}(\epsilon)\in \mathbb{N}$. 
%Similarly the second,  using (\ref{superexp}), remarking that $h^{(n_{l_0})}/  h^{(n_{l_{-1}})} \leq \| \ZLp{n_{l_{-1}},n_{l_0} } \|$ and using Lemma \ref{logintegrabilitylemma}.

Combining (\ref{logsumisepsilon}) and (\ref{lambdavsh}) to estimate the LHS of (\ref{mainwlambda}) from below,
% and then using it to get a lower estimate of the LHS of (\ref{mainwlambda}), we obtain
%\bes
% \frac{(1-\epsilon) }{e^{2\epsilon}}   h^{(n_{l_0})}_{j_0}   \log \frac{1}{\lambda^{(n_{l_{-2}})}}  \geq 
% \frac{(1-\epsilon) }{\nu}   h^{(n_{l_0})}_{j_0}   \log \frac{1}{\lambda^{(n_{l_0})}}  (1-\epsilon)
%\ees
we get the lower estimate that complete the proof of the Lemma.
%which is the lower estimate that complete the proof of (\ref{estimatemain}) in the Lemma.
\end{proof}

\paragraph{Estimate of the Gap Error.}\label{estimategapsec}
%We will now show that the gap error in (\ref{BSdecomp}) is negligible compared to the main order term estimated in Lemma \ref{estimatemainlemma}.
%More precisely, let us prove the following. 
\begin{lemma}[Gap error.]\label{estimategaplemma}
For each $\epsilon$, if $l_0 > l_{g}(\epsilon)$,
\be \label{estimategap}
0 \leq  \sum_{x_i \in I^{(n_{l_{-2}})} \backslash  I^{(n_{l_0})}}   \frac{1}{x_i}\,  \leq
 \epsilon    
\left( h^{(n_{l_0})}_{j_0}  \log h^{(n_{l_0})} \right).
%\left( \frac{1}{ \lambda^{(n_{l_0})} }  \log \frac{1}{\lambda^{(n_{l_0})}} \right).
\ee 
\end{lemma}
\begin{proof}
The bound below is trivial since $1/x_i >0$. Since we are considering $0\leq i < r_0=h^{(n_{l_0})}_{j_0}$, it follows from the tower construction (see Section \ref{towersec}) that the points $x_i= T^ix_0$ of the orbit of $x_0\in I^{(n_{l_0})}_{j_0}$ belong to different floors of the tower $Z^{(n_{l_0})}_{j_0}$ and that their minimum distance is bounded from below by
\bes
\min_{0\leq i,j < r_0} |x_i - x_j | \geq  
\lambda^{(n_{l_0})}_{j_0} 
\geq  
\frac{1}{d \kappa \nu h^{(n_{l_0})}_{j_0}} ,
%\frac{1}{d\nu} \lambda^{(n_{l_0})} ,
\ees
where in the last inequality we used that $n_{l_0}$ is balanced and Lemma \ref{balancehlambda}.

Remarking also that $x_0$ is the closest point to the singularity, it follows that if we rearrange the $x_i$ in increasing order and relabel them $\tilde{x}_i$ ($\tilde{x}_{i}< \tilde{x}_{i+1}$), we have 
\bes
\tilde{x}_i \geq x_0 + 
%i \lambda^{(n_{l_0})}_{j_0} 
\frac{i}{d \kappa \nu h^{(n_{l_0})}_{j_0}} 
%\lambda^{(n_{l_0})} 
, \qquad i=0, \dots , r_0-1.
\ees
%Moreover $x_i $ can be rearranged 

Since the roof function $1/x$ is monotonically decreasing, the gap error can be bounded from above by
\be \label{arithmeticbound}
 \sum_{x_i \in I^{(n_{l_{-2}})} \backslash  I^{(n_{l_0})}}   \frac{1}{x_i}\,  \leq     \sum_{k=1}^{K}  \frac{1}{x_0 + k 
%\frac{1}{d\nu} 
/ d \kappa \nu h_{j_0}^{(n_{l_0})}}
%\lambda^{(n_{l_0})}}_{j_0} 
,
\ee 
where $K=\# \{ x_i \in  I^{(n_{l_{-2}})} \backslash  I^{(n_{l_0})} \} $ and $k\geq 1$ since $x_0 \in  I^{(n_{l_0})}$.

%In order to estimate the RHS, i.e. the contribution given by the worst hypothetic case of an arithmetic progression, we can use 
The following lemma is proved by Kochergin in \cite{Ko:nonI} as Lemma 5.1.
\begin{lemma}\label{Kocharithmetic} Let $h>0$ and $x>0$.
%[Kochergin]
\bes
\sum_{k=1}^{K} \frac{1}{x+k h} = \frac{1}{h} \log \left( \frac{t_0 + K}{t_0+1} \right) + \frac{1}{h} R_K(t_0) ,
\ees
where $t_0=x/h$ and $0 <  R_K(t_0) < 1/(t_0+1)$.
\end{lemma}
%Since $\log \left( \frac{t_0 + K}{t_0+1} \right)$ is decreasing in $t_0$, it assumes its maximum at $t_0=0$ and moreover $R_K\leq 1$. Hence
%\bes
%\sum_{k=1}^{K} \frac{1}{x+k h} = \frac{1}{h} (\log ( {K} ) + 1).
%\ees 
Applying Lemma \ref{Kocharithmetic} to (\ref{arithmeticbound}) and using that $\log \left( \frac{t_0 + K}{t_0+1} \right)$ is decreasing in $t_0$, so it reaches its maximum $\log K$ at $t_0=0$, 
% and moreover $R_K\leq 1$  it gives
\be \label{arithmetic}
 \sum_{x_i \in I^{(n_{l_{-2}})} \backslash  I^{(n_{l_0})}}   \frac{1}{x_i}\, \leq 
 { d\kappa \nu}{h^{(n_{l_0})}_{j_0}} (\log {K}  + 1). 
\ee 
The cardinality $K$ of points $x_i \in  I^{(n_{l_{-2}})} \backslash  I^{(n_{l_0})} $ can be bounded by Remark \ref{dynmean} in terms of the cocycle matrices, by
\be \label{K}
K \leq \sum_{j=1}^{d} \ZLp{n_{l_{-2}},n_{l_0}}_{j j_0} \leq \| \ZLp{n_{l_{-2}},n_{l_0}} \| .
%\leq \prod_{l=l_{-2}}^{l_{-2}+L-1} \| \ZLp{n_{l},n_{l+1}} \|.
\ee
%Using the property (\ref{integrabiliy}) of balanced times and reasoning in the same way than at the end of Lemma \ref{estimatemainlemma} (see especially (\ref{logsumisepsilon}))
Hence, applying (\ref{arithmetic}) and (\ref{K}) we get
\be \label{gapratio}
\frac{ \sum_{x_i \in I^{(n_{l_{-2}})} \backslash  I^{(n_{l_0})}}   \frac{1}{x_i}  }{  { h^{(n_{l_0})}_{j_0} }  \log {h^{(n_{l_0})}}  } \leq  d \kappa \nu  \frac{ \log {K}  + 1} {\log  {h^{(n_{l_0})}} } \leq   
 d \kappa \nu  \frac{ \log {\| \ZLp{n_{l_0 -{(L_1 + L_2)\overline{l}}},n_{l_0}} \|   }  + 1} {\log  {h^{(n_{l_0})}} }.
%\frac{d\nu   \sum_{l=l_{-2}}^{l_{-2}+L-1}\log \| \ZLp{n_{l},n_{l+1}}\|}{ l_0 \log d } + \frac{d\nu  }{\log{\frac{1}{\lambda^{(n_{l_0})}}}} .
\ee
The RHS can be made smaller than $\epsilon$ using again Lemma \ref{loglimit} and Corollary \ref{superexp} as long as $l_0\geq l_g$ for some $l_g(\epsilon)\in \mathbb{N}$.
%where to estimate the denominator we used (\ref{logdbound}). Clearly the very last term in the LHS of (\ref{gapratio}) can be made smaller than $\epsilon/2 d\nu$ if $r_0\gg1$. Using the property (\ref{integrability}) of balanced times and
%%%%%%%%%%%%%%%%%USE A SEPARATE LEMMA FOR (\ref{logsumisepsilon}),
 %reasoning in the same way than at the end of Lemma \ref{estimatemainlemma} to estimate (\ref{logsumisepsilon}), there exists some $R_g>0$ such that if $r\geq R_g$ also the first term is smaller than $\epsilon/2 d\nu$. This gives (\ref{estimategap}) and completes the proof of the Lemma.
\end{proof}

%%%%ACCORCIABILE
Recalling the decomposition (\ref{BSdecomp}) of the Birkhoff sums, the estimates of the main contribution and of the gap error in Lemma \ref{estimatemainlemma} and Lemma \ref{estimategaplemma} combined together, for $l_0\geq$ $l(\epsilon)\doteqdot  $$ \max\{ l_m, l_g\}$, yield the estimate in Proposition \ref{alongtower}. 
%%%
%choosing $\epsilon$ accordingly to $\varepsilon$,  for $l_0 \geg \doteqdot max \{ l_m, l_g, \overline{l} (1+L_1+L_2)\}$, the estimates of the main contribution and of the gap error in Lemma \ref{estimatemainlemma} and Lemma \ref{estimategaplemma} respectively, combined together yield the estimate in Proposition \ref{alongtower}. 
\end{proofof}

\subsubsection{Growth of Birkhoff sums for other points.}\label{generalx}
In this section we get an estimate for $\BS{u}{r}(x)$ using the estimate 
% for the Birkhoff sum along a tower
found in Section \ref{BSalongtowersec} as a fundamental block, i.e. decomposing $\BS{u}{r}(x)$ into pieces which correspond to Birkhoff sums along a tower. It turns out that singular errors from points in the bottom floors of the towers (see the terminology introduced just after (\ref{BSdecomp})) could prevent from getting an estimate of order $r\log r$. In order to get this type of asymptotic, it is necessary to throw away a set of initial points $x$ which has an arbitrarily small measure. 
The integrability condition (\ref{integrability}) of the sequence of balanced times is used in its full strength only in this part.

%%%ELIMINABILE
%In Proposition \ref{BSgeneralxprop} we state a condition on $x$ which assures the right asymptotic growth. When constructing the mixing partition in Section \ref{mixingpartitionsec}, we will show that it is possible to exclude for each given $t$, a set of points of measure tending to zero in $t$ so that for the remaining set of initial points $x$ it is possible to control at the same time all sums of the form $\BS{f}{r(x,t)}(x)$ by simultaneously verifying the assumptions of  Proposition \ref{BSgeneralxprop}.
%%%

% excluding, for each given $t$, a set of points of measure tending to zero in $t$. Moreover for the remaining set of inital points $x$ it is possible to control at the same time all sums of the form $\BS{f}{r(x,t)}(x)$.
%The key property used to achieve the estimates in  Proposition \ref{BSgeneralxprop} is 
%%%%O DOPO
%The integrability condition (\ref{integrability}) of the sequence of balanced times %constructed in Section \ref{existencebalancedtimessec},which 
%is used in its full strength only in this part.
%It plays an analogouse role of the diophantine conditions for the rotation angle in \cite{SK:mix, Ko:nonI}.

%\paragraph{Notation for approximation by towers.}
%Denote by $\Orb{x}{r}$ the segment of orbit $\{ T^i x$, $0\leq i < r \}$; Consider
% a tower $Z^{(n)}_j$.
%Then, in what follows, for convenience, we write 
%\begin{displaymath} 
%\Orb{x}{r} \prec Z^{(n)}_j \quad \mathrm{iff} \quad \exists \,k  \,| \quad 0\leq k \leq h_j^{(n)}-r , \quad 
% T^i x \in  T^{k+i}  I^{(n)}_j 
%, \quad  0\leq i < r.
%\end{displaymath}
%When $\Orb{x}{r} \prec Z^{(n)}_j$, the orbit segment $\Orb{x}{r}$ is contained in $Z^{(n)}_j$, each point of $\Orb{x}{r}$ is contained in a different floors of $Z^{(n)}_j$ and $T$ acts on the orbit points by shifting them to the next floor. Visually, $\Orb{x}{r} \prec Z^{(n)}_j$ is represented in Figure \ref{OinZ}, where for convenience the tower $Z^{(n)}_j$ is drawn horizontally.

%Assume $\Orb{x}{r} \prec Z^{(n)}_j$. 
%We write $\Orb{x}{r} \prec Z_1 \wedge Z_2$, where $Z_1, Z_2 \in \xi_n (Z^{(m)}_j)$, $n<m$  (see Section \ref{towersec}), if $\Orb{x}{r} \subset Z_1 \cup Z_2$ and $\Orb{x}{r} \cap Z_1 \cap Z_2 \neq \emptyset$. See Figure \ref{OinZ1Z2}. 
%More generally, we write $\Orb{x}{r} \prec Z_1 \wedge Z_2$, where $Z_1, Z_2 \in \xi_n (Z^{(m)}_j)$, $n<m$ and $Z_1\subset Z^{(n)}_j_1$, $Z_1\subset Z^{(n)}_j_2$,  if there exists $ 0\leq k \leq h_{j_1}^{(n)} $ such that $T^i x \in  T^{k+i} \left( I^{(n)}_{j_1} \cap Z_1 \right)$ for $0\leq i < h_{j_1}^{(n)} - k $ and $ T^{ h_{j_1}^{(n)} - k  +i}x \in T^i \left( I^{(n)}_{j_1} \cap Z \right)$ for $0\leq i \leq $. As can shown in Figure \ref{OinZ1Z2}, the orbit runs inside two adiacent towers $Z_1, Z_2 \in \xi_n (Z^{(m)}_j)$.

%On the other side, given $ Z \in \xi_n (Z^{(m)}_j)$ with $Z\subset Z^{(n)}_i$, we write
%\begin{displaymath} 
%Z \prec \Orb{x}{r}  \quad \mathrm{iff} \quad \exists \, 0\leq k \leq r- h_i^{(n)} \,| \quad 
% T^{i+k} x \in  T^{i} \left( I^{(n)}_j \cap Z \right), \quad  0\leq i < h_i^{(n)} .
%\end{displaymath}
%When $Z  \prec \Orb{x}{r}$, there is exactly one point of $\Orb{x}{r}$ in each floor of $Z$ and again  $T$ acts on these points by shifting them to the next floor. Visually, $Z \prec \Orb{x}{r} $ is represented in Figure \ref{ZinO}.

%More generally, $Z_1 \wedge Z_2 \prec \Orb{x}{r}$, where  $Z_i \subset  Z^{(n)}_{j_s}$ for $s=1,2$,  if both $Z_1  \prec \Orb{x}{r}$ and $Z_2  \prec \Orb{x}{r}$ and moreover  $Z_1$ and  $Z_2$ are  neighbouring elements of $\xi_n (Z^{(m)}_j)$, meaning that $T^{h^{(n)}_{j_1}}( I^{(n)}_{j_1} \cap Z_1) =  I^{(n)}_{j_2} \cap Z_2$ (see Figure \ref{Z1Z2inO}). The notation extends to $Z_1 \wedge \dots \wedge Z_N \prec \Orb{x}{r}$.

\paragraph{Preliminary notation.}
Let $\{ n_l \}_{l \in \mathbb{N}} $ be the sequence of balanced times in Proposition \ref{existencebalancedtimes}.
%, which depends on the IET $T$ we start with.
Assume $h^{(n_l)} \leq r < h^{(n_{l+1})}$. 
%We will throw away different sets of points to estimate $\BS{u}{r}$ $h^{n_l} < r < h^{n_l}$. 
% where 
%The following auxiliary quantities $N$, $L$, $n_{\overline{l}}$ and $\sigma_r$ will be used in formulating Proposition \ref{BSgeneralxprop}. 
%Given $\epsilon>0$, let 
%$N=N(\epsilon)$ be such that 
%\be \label{Nepsilondef}
%\min \left\{ \left| \frac{d^N +2}{d^N} -1 \right| ,  \left| \frac{d^N}{d^N+2 } -1 \right| \right\} < \epsilon .
%\ee
%Assume we can associate to $r \in \mathbb{N}$ and $x\in I^{(0)}$ a balanced Zorich time $n_{\overline{l}}$, where  ${\overline{l}}= {\overline{l}}(r)$ is a function of $r$, such that $Z \prec  \Orb{x}{r} \prec Z^{(n_{\overline{l}})}_j $, for some $1\leq j\leq d$, where $Z \in  \xi_{n_{\overline{l}}} (Z^{(n_{\overline{l}-L})}_{i})$ for $1\leq i\leq d$, (see also Figure \ref{towersinclusion}) and $L  \in \mathbb{N} $ is independent on $r$. In particular $h_i^{(\overline{l}-L)} < r < h_j^{(\overline{l})}$. 
%In section \ref{} we will construct for each given $t$ a set with large measure of initial points $x$ such that $n_{\overline{l}}(r)$ and $L$ with the above properties exist for all $r=r(x,t)$.
Define the following sequence $\{ \sigma_l \}_{l\in \mathbb{N}}$, used in the proof of Proposition \ref{BSgeneralxprop} below as a threshold to determine whether $r$ is closer to $h^{(n_l)}$ or to $h^{(n_{l+1})}$.
%the sequence $\{ \sigma_l \}_{l \in mathbb{N}}$
% as follows.%\footnote{The notation $\sigma_l$ is chosen in analogy to \cite{Ko:nonI}.}
Let $\tau'$ be such that $\tau/2 < \tau' < 1$, where $\tau$ is the diophantine exponent in (\ref{integrability}) given by Proposition \ref{existencebalancedtimes} and $\tau '$ is well defined since $\tau <2$. Let
%Clearly $\sigma_l$ depends on the IET $T$ we start with, since the sequence $\{n_l\}_{l\in \mathbb{N}}$ does.
\be \label{sigmadef}
\sigma_l = \sigma_l (T) \doteqdot \left( \frac{ \log \| \ZLp{n_{l} , n_{l+1} } \|} {\log h^{(n_l)}}  \right)^{\tau'} ,  \qquad \frac{\tau}{2} < \tau' < 1,
%\sigma_r \doteqdot \max \left\{ \sqrt{ \frac{1}{\log \log r} },\, \sqrt{\frac{\log \| \ZLp{n_{\overline{l}-L-N} , n_{\overline{l}} } \|}{\log r} } \right\} .
\ee
Clearly $\sigma_l$ depends on the IET $T$ we start with, since the sequence $\{n_l\}_{l\in \mathbb{N}}$ does.
%where $\tau$ is the diophantine exponent in (\ref{integrability}) given by Proposition \ref{existencebalancedtimes} and $\tau '$ is well defined since $\tau <2$.
%The sequence  $\{ \sigma_l \}$ is used in the proof of Proposition \ref{BSgeneralxprop} below as a threshold to determine whether $r$ is closer to $h^{(n_l)}$ or to $h^{(n_{l+1})}$.% (see the proof of Proposition \ref{BSgeneralxprop} below). We state in a separate Lemma some of its easy properties needed later.

\begin{lemma}\label{sigmarlemma}
The sequence $\{ \sigma_l \}_{l \in \mathbb{N}}$ satisfies the following properties:
\begin{itemize}
\item[(i)] 
$\lim_{l\rightarrow +\infty} \sigma_l =0 $;
%\item[(ii)] $\lim_{l \rightarrow +\infty} \sigma_l  \log h^{(n_l)}  = + \infty $;
\item[(ii)] $ \lim_{l\rightarrow +\infty} \left( \frac{\log \| \ZLp{n_{l} , n_{l+1} } \|}{\log h^{(n_l)} }  \right) \frac{1}{ \sigma_l} =0 $;
\item[(iii)] $\lim_{l \rightarrow +\infty} \sigma_l  \log h^{(n_l)}  = + \infty $;
\item[(iv)] $ \lim_{l\rightarrow +\infty} \sigma_l ^2  h^{(n_{l+1})} \lambda^{(n_l)}  =0$.
%\left( \frac{\log \| \ZLp{n_{l} , n_{l+1} } \|}{\log h^{(n_l)} }  \right) / \sigma_l =0 $;
%\item[(i)] 
%$\lim_{r\rightarrow +\infty} \sigma_r =0 $;
%\item[(ii)] $\lim_{r\rightarrow +\infty} \sigma_r ^2  \log r= \lim_{r\rightarrow +\infty} \sigma_r   \log r  = + \infty $;
%\item[(iii)] $ \lim_{r\rightarrow +\infty} \left( \frac{\log \| \ZLp{n_{\overline{l}-L-N} , n_{\overline{l}} } \|}{\log r}  \right) / \sigma_r =0 $.
\end{itemize}
\end{lemma}

%%%%%ELIMINABILE
\begin{proof}
Both $(i)$ and $(ii)$ follow from Lemma \ref{loglimit}. % remarking respectively that $\tau'>0$ and $1-\tau'>0$.
 %In order to show $(i)$, since $\lim_{r\rightarrow +\infty} 1/\log \log r =0 $, it is enough to show that also the second expression in the LHS of (\ref{sigmadef}) limits to zero. Remark that $r > h_i^{(n_{\overline{l}-L-N})} $ and by using the heights balance (\ref{heightsbalance}) (see Lemma \ref{balancehlambda}, equation (\ref{hcomparedtolambda}))
%% and (\ref{lengthsbalance}))
% and  (\ref{logdbound}), 
%\be\label{logrbound}
%\log r > \log h_i^{(n_{\overline{l}-L-N})} >  \log \frac{1}{\nu  \lambda^{(n_{\overline{l}-L-N})}}  > (\overline{l}-L-N) \log d - \log \nu > \frac{1}{2}(\overline{l}-L-N) \log d ,
%\ee
%where the last inequality holds if $\overline{l} > L + 2 \log \nu /\log d $. Clearly $\overline{l} \gg 1$ can be achieved choosing $r \gg 1$, 
% since  $l \gg 1$, which can be clearly achieved if $r \gg 1$, 
%because $r <  h_j^{(\overline{l})} $ and hence for any $l_0>0$, we get $\overline{l}>l_0$ as long as $r> h^{({l_o})}$.
%Using (\ref{logrbound}) and the property (\ref{integrability}) of balanced times and
% reasoning again as to estimate (\ref{logsumisepsilon}) at the end of Lemma \ref{estimatemainlemma}, 
%\be 
%\frac{\log \| \ZLp{n_{\overline{l}-L-N} , n_{\overline{l}} } \|}{\log r}  \leq  2 \frac{\sum_{l=\overline{l}-L-N}^{\overline{l}-1} \log \left\| \ZLp{n_{l} , n_{l+1} } \right\|}{ (\overline{l}-L-N) \log d   }  \leq \epsilon ,
%\ee
%as long as $r$ is sufficiently large. This completes the proof of $(i)$.
To show $(iii)$, remark that $ \log \| \ZLp{n_{l} , n_{l+1} } \| \geq \log d \geq 1$, so $\sigma_l \log h^{(n_l)} \geq (\log h^{(n_l)}) ^{1-\tau'} $ and apply Corollary \ref{superexp}.
%%% and since $\tau'<1$, $(ii)$ follows by Corollary \ref{superexp}.
%\bes
%\lim_{r\rightarrow +\infty} \sigma_r^2  \log r \geq \lim_{r\rightarrow +\infty}   \frac{\log r}{\log \log r}  = + \infty.
%\ees
%The second limit in $(ii)$ is a consequence of $(i)$.
For $(iv)$, using in order balance (see Lemma \ref{balancehlambda}), the transformation relation for heights, the definition of $\sigma_l$, the diophantine Property (\ref{integrability}) in Proposition \ref{existencebalancedtimes} and Corollary \ref{superexp}, we get 
\bes
\begin{split}
% \lim_{l\rightarrow +\infty} 
&\sigma_l ^2  h^{(n_{l+1})} \lambda^{(n_l)}  \leq \sigma_l ^2 \kappa \frac{h^{(n_{l+1})}}{h^{(n_l)}} \leq  \sigma_l ^2 \kappa   \| \ZLp{n_{l} , n_{l+1} } \| \leq \\
%\leq \kappa  \frac{( \log \| \ZLp{n_{l} , n_{l+1} } \| )^{2\tau'} \| \ZLp{n_{l} , n_{l+1} } \| }{ (\log h^{(n_l)})^{2\tau'}}
%\ees
%Recalling the definition of $\sigma_l$ and using the diophantine Property (\ref{integrability}) in Proposition \ref{existencebalancedtimes} and Corollary \ref{superexp}, we get
%\bes
%\begin{split}
%0 \leq  \lim_{l\rightarrow +\infty} \sigma_l ^2  h^{(n_{l+1})} \lambda^{(n_l)} 
%& 
& \leq   \lim_{l\rightarrow +\infty}   \frac{\kappa ( \log \| \ZLp{n_{l} , n_{l+1} } \| )^{2\tau'}  \| \ZLp{n_{l} , n_{l+1} } \| } { (\log h^{(n_l)})^ {2\tau'}} \leq
%\\ & \leq 
  \lim_{l\rightarrow +\infty} 
const \frac{ (\log l) ^{2\tau'} o( l^{\tau} )}{l^{2 \tau ' }} =0,
   \end{split}
\ees
where the last limit is zero since $2\tau'>\tau$.
\end{proof}

%Again using the definition of  $\sigma_r$ also $(iii)$ follows trivially from property $(i)$:
%%%%%%%%%%SHORTABLE
%\bes
%\lim_{r\rightarrow +\infty} \frac{\frac{\log \| \ZLp{n_{\overline{l}-L-N} , n_{\overline{l}} } \|}{\log r}  )}{ \sigma_r}  \leq 
%%\lim_{r\rightarrow +\infty} \frac{\frac{\log \| \ZLp{n_{\overline{l}-L-N} , n_{\overline{l}} } \|}{\log r}  )}{ \sqrt{\frac{\log \| \ZLp{n_{\overline{l}-L-N} , n_{\overline{l}} } \|}{\log r} } } = 
%\lim_{r\rightarrow +\infty}\sqrt{\frac{\log \| \ZLp{n_{\overline{l}-L-N} , n_{\overline{l}} } \|}{\log r} } =0.
%\ees
%\end{proof}

%To simplify the notation we will also define 
%\bes
%\overline{l}' \doteqdot \overline{l} - L - N(\epsilon) .
%\ees

%\paragraph{Growth of Birkhoff sums.}

%Consider the following set, where $[\cdot]$ denotes the fractional part.
\begin{defn} Let $\Sigma^+_{l}=\Sigma^+_l (T)$ be the following set, where $[\cdot]$ denotes the fractional part:
%Consider the following set, where $[\cdot]$ denotes the fractional part.
\be \label{badsetforBSu}
%\Sigma_{l}=
\Sigma^+_l (T) 
\doteqdot \bigcup_{i=0}^{[\sigma_l h^{(n_{l+1})}]} T^{-i} [0, \sigma_l \lambda^{(n_l)}].
\ee
\end{defn}
\noindent Remark that, by Property $(iv)$ of Lemma \ref{sigmarlemma},
%$Leb(\Sigma^+_l) \xrightarrow{l\rightarrow +\infty} 0$
\be\label{LebSigma}
Leb(\Sigma^+_l) \leq (\sigma_l h^{(n_{l+1})})( \sigma_l \lambda^{(n_l)}) \xrightarrow{l\rightarrow +\infty} 0.
\ee
%by Property $(iv)$ of Lemma \ref{sigmarlemma}.

%Let $x_m \doteqdot \min_{0\leq i < r} T^i x$. 
%Using the notation introduced in the previous paragraph, we can formulate the following.
\begin{prop}[Growth of Birkhoff sums for general points]\label{BSgeneralxprop} Let $T\in \M^+$.
For any $\varepsilon > 0 $ there exists $l_o >0$ such that for $l\geq l_o$, then for each $r \in \mathbb{N}$ and $x\in I^{(0)}$ such that 
\begin{eqnarray} \label{assumptiongeneralxprop}
h^{(n_l)}\leq r < h^{(n_{l+1})} \, &\mathrm{\it and}& \, x \notin  \Sigma^+_{l}(T) , \\%\qquad \mathrm{\it we\, get}\\
 \label{BSgeneralxestimate}
(1-\varepsilon )r\log r \leq & \BS{u}{r}(x) & \leq 
 (1+\varepsilon ) r\log r + \frac{\kappa +1}{x_m},
\end{eqnarray}
%\be \label{assumptiongeneralxprop}
%h^{(n_l)}\leq r < h^{(n_{l+1})} \quad \mathrm{\it and} \quad x \notin  \Sigma^+_{l}(T) , \quad \mathrm{\it we\, get}
%\ee 
%denoting by $x_m \doteqdot \min_{0\leq i < r} T^i x$, we get \footnote{The constant $\kappa$ is the same given by Proposition \ref{existencebalancedtimes}.}
%\be \label{BSgeneralxestimate}
%(1-\varepsilon )r\log r \leq  \BS{u}{r}(x)  \leq 
% (1+\varepsilon ) r\log r + \frac{\kappa +1}{x_m},
%\ee
where $x_m \doteqdot \min_{0\leq i < r} T^i x$ and $\kappa$ is given by Proposition \ref{existencebalancedtimes}.
%, we get \footnote{The constant $\kappa$ is the same given by Proposition \ref{existencebalancedtimes}.}
%where $\kappa$ is given by Proposition \ref{existencebalancedtimes}.
%Assume $\nu>1$, $\epsilon>0$ are given, $N(\epsilon)$ satisfies (\ref{Nepsilondef}) and $L\in \mathbb{N}$. Consider $r$ and $x$ 
%for which there exists $\overline{l}=\overline{l}(r)$ such that 
%\begin{itemize}
%\item[(a)] $\overline{Z} \prec  \Orb{x}{r} \prec Z^{(n_{\overline{l}})}_{j_0} $ for some $1\leq j_0 \leq d$, where $\overline{Z} \in  \xi_{n_{\overline{l}-L}} (Z^{(n_{\overline{l}})}_{j_0})$. 
%\item[(b)] 
%$x \notin \cup _{k=0}^{h^{(n_{\overline{l} - L -N})}} T^{-k} \left[ 0, 2 \nu^2 d \sigma_r \lambda_1^{(n_{\overline{l} - L -N})} \right] $, where $ \sigma_r=(\sigma_r (L, \epsilon, N)$ was defined in (\ref{sigmadef}). 
%\end{itemize}
%Then there exists $R$ such that if $r\geq R$
%\be
%\frac{(1-\epsilon)^2}{\nu} 1 \leq  \frac{\BS{1u}{r}(x)}{r \log r} \leq 
% \nu (1+2 \epsilon) 1.
%\ee
\end{prop}
By adding a small measure set to the excluded set $\Sigma^+_l$ of initial points,  as in Section \ref{mixingpartitionsec}, one can take into account also the term $\kappa/x_m$ and get the asymptotic $r\log r$ for $\BS{u}{r}(x)$.

The following notation is used in the proof of Proposition \ref{BSgeneralxprop}.
\paragraph{Notation for approximation by towers.}
%\paragraph{Preliminary Notation for Proposition \ref{BSgeneralxprop}.}
%For convenience, let us introduce the following notation. 
Denote by $\Orb{x}{r}$ the orbit segment $\{ T^i x$, $0\leq i < r \}$. Consider a tower $Z^{(m)}_j$. 
% or a piece of a tower $Z\subset Z^{(m)}_j$, $Z\in \xi_n (Z^{(m)}_j)$, $n<m$ (see Section \ref{towersec}). 
In what follows, we write 
%\begin{defn} We write $\Orb{x}{r} \prec Z$ if and only if there exists $0\leq k \leq h_j^{(n)}-r $ such that 
\begin{displaymath} 
\Orb{x}{r} \prec Z^{(m)}_j \quad \mathrm{iff} \quad \exists \,k  \,| \quad 0\leq k \leq h_j^{(m)}-r , \quad 
 T^i x \in  T^{k+i}  I^{(m)}_j 
%\cap Z 
, \quad  0\leq i < r.
\end{displaymath}
% \end{defn}
When $\Orb{x}{r} \prec Z^{(m)}_j$, 
% the orbit segment $\Orb{x}{r}$ is contained in $Z^{(m)}_j$, 
each point of $\Orb{x}{r}$ is contained in a different floor of $Z^{(m)}_j$ and $T$ acts on the orbit points $T^ix$ ($0\leq i < r-1$) by shifting them to the next floor. (see Figure \ref{OinZ}, where the tower $Z^{(m)}_j$ is drawn horizontally.)
%Visually, $\Orb{x}{r} \prec Z^{(m)}_j$ is represented in Figure \ref{OinZ}, where for convenience the tower $Z^{(m)}_j$ is drawn horizontally.
%%%ELIMINABILE togli anche ref alla figure!!
\begin{figure}\label{OinZ}
\centering
\subfigure[$\Orb{x}{r} \prec Z_1\wedge \dots \wedge Z_4$, ]{\label{OinZ1Z4}{\includegraphics[width=0.45\textwidth]{OinZ1Z4.eps}}}
\hspace{5mm}
\subfigure[$ Z_1\wedge Z_2 \prec \Orb{x}{r} $]{\label{Z1Z2inO}{\includegraphics[width=0.45\textwidth]{Z1Z2inO.eps}}}
\caption{$\Orb{x}{r} \prec Z^{(m)}_j$  }
%\caption{One step of Rauzy-Veech algorithm on $T=(\lambda, (4321))$.}
\end{figure}



Assume $\Orb{x}{r} \prec Z^{(m)}_j$. 
We write $\Orb{x}{r} \prec Z_1 \wedge Z_2 \wedge \dots \wedge Z_N$, where $Z_i \in \xi_n (Z^{(m)}_j)$, $n\leq m$ (see Section \ref{partitionsec}) for $i=1, \dots, N$, if $\Orb{x}{r} \subset  \cup_{i=1}^{N} Z_i$, $\Orb{x}{r} \cap Z_i \neq \emptyset$ for $1\leq i \leq N$ and moreover $Z_i$ are \emph{consecutive} partition elements of $\xi_n (Z^{(m)}_j)$, i.e., if $h_i$ denotes the height of $Z_i$, remarking that $Z_i\cap I^{(n)}$ is the base of $Z_i$, we have $T^{h_i} Z_i \cap I^{(n)} = Z_{i+1} \cap  I^{(n)}$ for $i=1,\dots, N-1$ (see e.g. Figure \ref{OinZ1Z4}). 
%More generally, we write $\Orb{x}{r} \prec Z_1 \wedge Z_2$, where $Z_1, Z_2 \in \xi_n (Z^{(m)}_j)$, $n<m$ and $Z_1\subset Z^{(n)}_j_1$, $Z_1\subset Z^{(n)}_j_2$,  if there exists $ 0\leq k \leq h_{j_1}^{(n)} $ such that $T^i x \in  T^{k+i} \left( I^{(n)}_{j_1} \cap Z_1 \right)$ for $0\leq i < h_{j_1}^{(n)} - k $ and $ T^{ h_{j_1}^{(n)} - k  +i}x \in T^i \left( I^{(n)}_{j_1} \cap Z \right)$ for $0\leq i \leq $. As can shown in Figure \ref{OinZ1Z2}, the orbit runs inside two adiacent towers $Z_1, Z_2 \in \xi_n (Z^{(m)}_j)$.
On the other hand, %given $ Z \in \xi_n (Z^{(m)}_j)$ with $Z\subset Z^{(n)}_i$, 
we write
 $Z_1 \wedge Z_2 \wedge \dots \wedge Z_N \prec \Orb{x}{r}$, where  $Z_i \in \xi_n (Z^{(m)}_j)$, $n<m$, if $\# \{ Z_i \cap \Orb{x}{r} \} = h_i$ for all $i=1\, \dots, N$, i.e. there is exactly one point of $\Orb{x}{r}$ in each floor of each $Z_i$ and moreover $Z_i$ are, as above, consecutive partition elements. (See Figure \ref{Z1Z2inO}.)

%  if both $Z_1  \prec \Orb{x}{r}$ and $Z_2  \prec \Orb{x}{r}$ and moreover  $Z_1$ and  $Z_2$ are  neighboring elements of $\xi_n (Z^{(m)}_j)$, meaning that $T^{h^{(n)}_{j_1}}( I^{(n)}_{j_1} \cap Z_1) =  I^{(n)}_{j_2} \cap Z_2$ (see Figure \ref{Z1Z2inO}). The notation extends to $Z_1 \wedge \dots \wedge Z_N \prec \Orb{x}{r}$.
%(visually, they are neighbouring elements of $\xi_n (Z^{(m)}_j)$).

%On the other side, given $ Z \in \xi_n (Z^{(m)}_j)$ with $Z\subset Z^{(n)}_i$, we write
%\begin{displaymath} 
%Z \prec \Orb{x}{r}  \quad \mathrm{iff} \quad \exists \, 0\leq k \leq r- h_i^{(n)} \,| \quad 
% T^{i+k} x \in  T^{i} \left( I^{(n)}_j \cap Z \right), \quad  0\leq i < h_i^{(n)} .
%\end{displaymath}
%When $Z  \prec \Orb{x}{r}$, there is exactly one point of $\Orb{x}{r}$ in each floor of $Z$ and again  $T$ acts on these points by shifting them to the next floor. Visually, $Z \prec \Orb{x}{r} $ is represented in Figure \ref{ZinO}.

%More generally, 
%$Z_1 \wedge Z_2 \prec \Orb{x}{r}$, where  $Z_i \subset  Z^{(n)}_{j_s}$ for $s=1,2$,  if both $Z_1  \prec \Orb{x}{r}$ and $Z_2  \prec \Orb{x}{r}$ and moreover  $Z_1$ and  $Z_2$ are  neighboring elements of $\xi_n (Z^{(m)}_j)$, meaning that $T^{h^{(n)}_{j_1}}( I^{(n)}_{j_1} \cap Z_1) =  I^{(n)}_{j_2} \cap Z_2$ (see Figure \ref{Z1Z2inO}). The notation extends to $Z_1 \wedge \dots \wedge Z_N \prec \Orb{x}{r}$.
%(visually, they are neighbouring elements of $\xi_n (Z^{(m)}_j)$).

\begin{proofof}{Proposition}{BSgeneralxprop} 
It is always possible to assume that $ \Orb{x}{r} \prec \overline{Z} \doteqdot Z^{(n_{\overline{l}})}_{\overline{j}}$ for some $\overline{l}\geq l+1$, since by choosing $\overline{l}$ such that $\lambda^{(n_{\overline{l}})} < x_m=\min_{0\leq i< r}T^ix$, we assure that $\Orb{x}{r} \cap I^{(n_{\overline{l}})} = \emptyset$.
%%%%%The idea of the proof is to cut the orbit $ \Orb{x}{r}$ into pieces which correspond to Birkhoff sums along the tower, so that we can apply to each of them Proposition \ref{alongtower}. Cutting it in many pieces allows to get an asymptotic estimate $r\log r$ from the main contributions,  
%from the two bounds in (\ref{alongtowerestimate}),
% but one has to consider separately the singular contributions from the points belonging to the bottom floors of each tower. These singular contributions could a priori spoil the asymptotic $r\log r$. The restriction on the set of initial points $x$ in assumption (\ref{assumptiongeneralxprop}) is essential in controlling them.
% these contributions.
\paragraph{Orbit decomposition into sums along towers.}
%It is always possible to assume that $ \Orb{x}{r} \prec \overline{Z} \doteqdot Z^{(n_{\overline{l}})}_{\overline{j}}$ for some $\overline{l}\geq l+1$, since by choosing $\overline{l}$ such that $\lambda^{(n_{\overline{l}})} < x_m=\min_{0\leq i< r}T^ix$, we assure that $\Orb{x}{r} \cap I^{(n_{\overline{l}})} = \emptyset$. We need $\overline{Z}$ only as a ``frame'' in which to
Let us approximate $\Orb{x}{r}$ with elements of $\xi_{n_{l+1}}(\overline{Z})$ and $\xi_{n_{l}}(\overline{Z})$.
Using the assumption $r<h^{(n_{l+1})}$, the cardinality $\# \Orb{x}{r} \cap  I^{(n_{{l+1}})}$ is bounded by $[\kappa]+1$: since the return time to $ I^{(n_{{l+1}})}$ is at least $\min_j{h^{(n_{l+1})}}\geq h^{(n_{l+1})}/\kappa$ by $\kappa$-balance of heights, 
there cannot be more than $r/ \min_j h_j^{(n_{l+1})} + 1 \leq [\kappa]+1 $ returns.
%either $\Orb{x}{r} \cap  I^{(n_{{l+1}})} = \emptyset$ or $\Orb{x}{r} \cap  I^{(n_{{l+1}})}= \{x_m\}$. Indeed if there were two points,   

%%%
Hence there exists $Z^{l+1}_0, Z^{l+1}_1, \dots, Z^{l+1}_{a^{l+1}+1} \in \xi_{n_{l+1}} ( \overline{Z})$, with $a^{l+1} \leq [\kappa]$ such that (see Figure \ref{orbitdecompfigure})
%%%%%ELIMINABILE con ref alla Figure!!!
\be \label{approxl+1}
\Orb{x}{r} \prec Z_0^{l+1} \wedge Z_1^{l+1} \wedge \dots \wedge Z_{a^{l+1}+1}^{l+1}.
\ee
\begin{figure}
\centering
%\subfigure[$\Orb{x}{r} \prec Z$]{\label{OinZ}{
\includegraphics[width=0.85\textwidth]{orbitdecomp.eps}
\hspace{5mm}
%\subfigure[$ \Orb{x}{r} \prec Z_1\wedge Z_2 $]{\label{OinZ1Z2}{\includegraphics[width=0.45\textwidth]{OinZ1Z2.eps}}}
\caption{Approximation of $ \Orb{x}{r}$ by elements of $\xi_{n_{l+1}}$, $\xi_{n_{l}}$ and $\xi_{n_{l-L}}$. }
\label{orbitdecompfigure}
\end{figure}
\noindent Approximating $ \Orb{x}{r} $ also with elements $Z^{l}_i \in \xi_{n_{l}} ( \overline{Z})$ (see again Figure \ref{orbitdecompfigure}):
\be \label{apprl}
 Z_1^{l} \wedge Z_2^{l} \wedge \dots \wedge Z_{a^{l}}^{l}  \prec \Orb{x}{r} \prec Z_0^{l} \wedge Z_1^{l}  \wedge \dots \wedge Z_{a^{l}}^{l} \wedge  Z_{a^{l}+1}^{l}.
\ee 
%where $a^{l} \leq ([\kappa]+1) 
%\max_j \# \xi_{n_{l+1}}(Z^{(n_{l+1})}_j ) \leq
% \| \ZLp {n_l, n_{l+1}}\|$, as can be seen by using (\ref{entriesBmn}) to relate the cardinality of the partion to the cocycle norms.
%
We still need another level of approximation. Let $L \doteqdot l_1 \overline{l} \in \mathbb{N}$ where $l_1$ is such that $2\kappa /d^{l_1} < \epsilon$. We can find elements  $Z^{l-L}_i \in \xi_{n_{l-L}} ( \overline{Z})$ such that (Figure \ref{orbitdecompfigure})
\be\label{apprl-L} 
 Z_1^{l-L} \wedge Z_2^{l-L} \wedge \dots \wedge Z_{a^{l-L}}^{l-L}  \prec \Orb{x}{r} \prec Z_0^{l-L} \wedge Z_1^{l-L}  \wedge \dots \wedge Z_{a^{l-L}}^{l-L} \wedge  Z_{a^{l-L}+1}^{l-L}.
\ee
%where $a^{l-L} \leq (a^{l}+2 )  \| \ZLp {n_{l-L}, n_{l}}\| $.
Denote by $h_i^{l}$ and $h_i^{l-L}$ the heights of $Z_i^{l}$ and $Z_i^{l-L}$ respectively. Remark that 
\be \label{restimate}
 \sum_{i=1}^{a^{l}} h_{i}^{l} \leq    \sum_{i=1}^{a^{l-L}} h_{i}^{l-L} \leq r \leq \sum_{i=0}^{a^{l-L}+1} h_{i}^{l-L} \leq \sum_{i=0}^{a^{l}+1} h_{i}^{l} .
\ee

%From the orbit decomposition (\ref{orbitdecomp}) and the following remark, 
%using that $i_{j+1}-i_{j} =  h^{(n_{\overline{l}-L-N})}_k$
%Using the heights balance (\ref{heightsbalance}), we also get an estimate for $r$:
%\be\label{restimate}
%\left(
% \frac{z}{\nu} 
%\right) 
% h^{(n_{\overline{l}-L-N})} \leq r \leq (z+2)   h^{(n_{\overline{l}-L-N})}
%\ee

%only one already contains it, while if some $T^ix \in  I^{(n_{{l+1}})}$,   
%By assumption $ \Orb{x}{r} \prec Z^{(n_{\overline{l}})}_{j_0} $. Consider the partition $\xi_{n_{\overline{l}-L-N}} (Z^{(n_{\overline{l}})}_{j_0})$. Let us truncate $\Orb{x}{r}$ into pieces which belong each to a different partition element.
%, in order to use the estimate in Proposition \ref{alongtower}) to each of them. 
%\paragraph{Orbit decomposition into sums along towers.}
%To determine those pieces, 

Let us truncate $\Orb{x}{r}$ into segments contained in different elements $Z^{l-L}_i \in \xi_{l-L}(\overline{Z})$. Remarking that $I^{(n_{l-L})}$ contains all the bases of the towers $Z_i^{l-L}$, let us denote 
% using visits to $I^{(n_{l-L})}$, which contains all the bases of the towers $Z_i^{l-L}$. Let
%Let $y^{l-L}_0 \doteqdot  \Orb{x}{r} \cap  Z_1^{l-L} \cap I^{(n_{l-L})} $. Let 
\be \label{defyil-L}
y^{l-L}_i \doteqdot  \Orb{x}{r} \cap  Z_i^{l-L} \cap I^{(n_{l-L})}; \quad i=1,\dots, a^{l-L}.
%y^{l-L}_{i+1} \doteqdot T^{(n_{l-L})}  y^{l-L}_{i}, \quad i=1,\dots  a^{l-L}+1.
\ee
Since $h^{l-L}_i$ is exactly the first return time of $y^{l-L}_{i}$ to $I^{(n_{l-L})}$, $y^{l-L}_{i+1} = T^{h^{l-L}_i}  y^{l-L}_{i} =  T^{(n_{l-L})}  y^{l-L}_{i}$. %since $h^{l-L}_i$ is exactly the first return time of $y^{l-L}_{i}$ to $I^{(n_{l-L})}$. 
Add also the two auxiliary points:
\be \label{defy0l-L}
y^{l-L}_0 \doteqdot (T^{(n_{l-L})})^{-1}  y^{l-L}_{1}; \qquad y^{l-L}_{ a^{l-L}+1} \doteqdot T^{(n_{l-L})}  y^{l-L}_{ a^{l-L}}.
\ee
From (\ref{apprl-L}), % we get the following orbits inclusions:%decompose the orbit as
%\bes
$ \bigcup_{i=1}^{a^{l-L}} \Orb{y_{i}^{l-L}}{h^{l-L}_{i}} \subset \Orb{x}{r} \subset \bigcup_{i=0}^{a^{l-L}+1} \Orb{y_{i}^{l-L}}{h^{l-L}_{i}}.$
%\ees
%Remark that $ y^{l-L}_{i} \in Z_i^{l-L} \cap I^{(n_{l-L})}$, i.e. $y^{l-L}_{i}$ are visits of $ \Orb{x}{k} $ to the bottom elements of the decomposition  (\ref{apprl-L}).
%Denote by $h_i^{l-L}$ the height of $Z_i^{l-L}$; hence $T^{(n_{l-L})}  y^{l-L}_{i} = T^{h^{l-L}_i}  y^{l-L}_{i}$.

%; also we can write $y^{l-L}_0 = T^k x$ for some $0\leq k \leq h^{(n_{l-L})}$ and that $T^{r-1}x = T^{k'} y^{l-L}_{a^{l-L}}$ for some $0\leq k' \leq h^{(n_{l-L})}$. 
%moreover $y^{l-L}_0 = T^k x$ for some $0\leq k \leq h^{(n_{l-L})}$ and 

%Consider visit to $I^{(n_{l-L})}$, letting
%\begin{eqnarray}
%i_0  & \doteqdot & \min \{i \,\, | \quad T^i x \in I^{(n_{l-L})}, \, 0\leq i <r \} , \nonumber \\
%i_{j+1}  & \doteqdot & \min \{i\, \, | \quad  i> i_{j}, \, T^i x \in I^{(n_{l-L})},\, 0\leq i <r \}, \nonumber 
%\\ z & \doteqdot& \# \{ i \,\, | \quad T^i x \in I^{(n_{l-L})}, \, 0\leq i <r  \}, \label{zdef} 
%\end{eqnarray}
%and denoting by $y^{l-L}_j  \doteqdot  T^{i_{j}}x$, where $0\leq j \leq z$. Clearly 
%If $y_j \in I^{(n_{l-L})}_k$ for some $1\leq k \leq d$, clearly $i_{j+1}-i_{j} =  h^{(n_{l-L})}_k$ and $\Orb{y_j}{i_{j+1}-i_{j}} \subset Z^{(n_{l-L})}_k$. 

%%%%%%%%
%Denoting by $h_j^{l-L}$ the height of $Z_j^{l-L}$, we can decompose the orbit as
%\be \label{orbitdecomp}
%\Orb{x}{r} = \Orb{x}{k} \cup \bigcup_{j=1}^{a^{l-L}-1} \Orb{y_j}{h_j^{(n_{l-L})}}  \cup  \Orb{y_{a^{l-L}-1}}{k'}.
%%\Orb{x}{r} = \Orb{x}{i_0} \cup \bigcup_{j=0}^{z-1} \Orb{y_j}{(i_{j+1}-i_{j})}  \cup  \Orb{y_z}{r-i_z}. 
%\ee 

%Remark that $y_j \in I^{(n_{l-L})}_k $ for some $1\leq k \leq d$, then $i_{j+1}-i_{j} =  h^{(n_{l-L})}_k$. Moreover $\Orb{y_j}{(i_{j+1}-i_{j})} \subset Z^{(n_{\overline{l}-L-N})}_k$, so each piece of the orbit is contained in a different partition element. Hence
%\be \label{upperboundz}
%z \leq \# \{ Z \in \xi_{n_{\overline{l}-L-N}} (Z^{(n_{\overline{l}})}_{j_0}) \} \leq \| \ZLp{n_{\overline{l}-L-N}, n_{\overline{l}}} \|.
%\ee
%To get also a lower bound, recall that by assumption there exists $\overline{Z} \in  \xi_{n_{\overline{l}-L}} (Z^{(n_{\overline{l}})}_{j_0})$ such that $\overline{Z} \prec  \Orb{x}{r}$, say $\overline{Z} \subset  Z^{n_{\overline{l}-L}}_{j_z}$. Hence, 
%we also have the lower bound
% recalling the by positivity (\ref{positive}) of balanced return times, we also have a lower bound 
%\be \label{lowerboundz}
%z \geq \# \{ Z \in \xi_{n_{\overline{l}-L-N}} (\overline{Z}) , \} =  \sum_{j}  \ZLp{n_{\overline{l}-L-N}, n_{\overline{l}-L }}_{jj_z} \geq d^N,
%\ee
%where we used the dynamical meaning of the cocycle entries (see \ref{dynamicalmeaning}) and the positivity (\ref{positive}) of balanced return times.

%From the orbit decomposition (\ref{orbitdecomp}) and the following remark, 
%%using that $i_{j+1}-i_{j} =  h^{(n_{\overline{l}-L-N})}_k$
%using the heights balance (\ref{heightsbalance}), we also get an estimate for $r$:
%\be\label{restimate}
%\left(
% \frac{z}{\nu} 
%\right) 
% h^{(n_{\overline{l}-L-N})} \leq r \leq (z+2)   h^{(n_{\overline{l}-L-N})}
%\ee

%We can rearrange $\BS{u}{r}(x)$ gathering together contributions coming from visits to different $Z^{l-L}_j$ and since  $1/x>0$ we get the following estimate for the Birkhoff sums:
As a consequence, since $u>0$, we get the following estimate for $\BS{u}{r}(x)$:
\be\label{BSgeneralxestimate2}
\sum_{i=1}^{a^{l-L}} \BS{u}{h_i^{l-L}}(y_i^{l-L}) \leq \BS{u}{r}(x) \leq \sum_{i=0}^{a^{l-L}+1} \BS{u}{h_i^{l-L}}(y_i^{l-L}) .
\ee
%where the auxiliary point $y_{-1}\doteqdot T^{- i_{-1}}x$, with (see again Figure \ref{yjposition})
%$i_{-1}$ minimum $i>0$ such that  $T^{-i} x \in I^{(n_{\overline{l}-L-N})}$. 
%\be \label{i-1def}
%i_{-1} \doteqdot \min \{i \,\, | \quad T^{-i} x \in I^{(n_{\overline{l}-L-N})}, \, 0\leq i <r \} .
%\ee
Each term in the summations in (\ref{BSgeneralxestimate2}) is a Birkhoff sum along a tower of step $n_{l-L}$. Hence we can apply Proposition \ref{alongtower} to each term and find $l_0\doteqdot l(\epsilon)+L$ such that for each $l\geq l_0$ we get
%Recalling that $h^{(n_{l-L})}/ \kappa \leq   h^{l-L}_i \leq h^{(n_{l-L})} $ by $\kappa$-balance of the heights, 
\begin{eqnarray}
\BS{u}{r}(x) &\geq & (1-\epsilon)
\sum_{i=1}^{a^{l-L}} h_{i}^{l-L}   \log {{ h^{(n_{{l}-L})}}} + \sum_{i=1}^{a^{l-L}} \frac{1}{y_i^{l-L}} \label{BSgenxbb} \\
\BS{u}{r}(x)  &\leq& (1+\epsilon)
\sum_{i=0}^{a^{l-L}+1} h_{i}^{l-L}   \log   h ^{(n_{{l}-L})}  
+  \sum_{i=0}^{a^{l-L}+1} \frac{1}{y_i^{l-L}}. \label{BSgenxba}
\end{eqnarray}
%\be
%z  \frac{(1-\epsilon)^2}{d \nu^2 } C   h^{(n_{\overline{l}-L-N})}  \ln  \frac{ h^{(n_{\overline{l}-L-N})}}{d \nu ^2 } + 
%\sum_{j=0}^{z-1} \frac{C}{y_j^{l-L}} \leq \BS{u}{r}(x) \leq
%\nu (1+2 \epsilon)\nu^2 C h ^{(n_{\overline{l}-L-N})} \log \left(  \nu^2  h ^{(n_{\overline{l}-L-N})} \right). 
%+  \sum_{j=-1}^{z} \frac{C}{y_j^{l-L}}
%\ee
Let us refer to the first term in the LHS of (\ref{BSgenxbb}) or (\ref{BSgenxba}) as \emph{ergodic term} and to the last term, i.e. the contributions of points in the bottom floors, as \emph{resonant term}\footnote{There is again an analogy with the terminology used by \cite{Ko:nonI, Ko:nonII}.}.
\paragraph{Ergodic term.}
%Let us show that the ergodic term itself gives the asymptotic growth in the Proposition. 
Taking the ratio of the ergodic term over $r\log r$ and applying the bounds (\ref{restimate}) for $r$:
\begin{eqnarray}
\frac{\left( \sum_{i=1}^{a^{l-L}} h_{i}^{l-L} \right)  \log  h^{(n_{{l}-L})}}{r\log r } \geq  \left( 1- \frac{2  h^{(n_{l-L})}}{r} \right) \frac{  \log  h^{(n_{{l}-L})}}{\log r }  \label{ergodicb} \\
\frac{\left( \sum_{i=0}^{a^{l-L}+1} h_{i}^{l-L}\right) \log  h^{(n_{{l}-L})}}{r\log r } \leq  \left( 1+ \frac{2  h^{(n_{l-L})}}{r} \right) \frac{  \log  h^{(n_{{l}-L})}}{\log r }  \label{ergodica}
%\frac{z  h^{(n_{\overline{l}-L-N})}  \log \frac{ h^{(n_{\overline{l}-L-N})}}{d \nu } }{r\log r} \geq  \frac{ \frac{z}{z+2}r \log \frac{r}{(z+2) d \nu}}{r\log r} \geq  \frac{z}{z+2} \left( 1- \frac{\log (z+2) d \nu }{\log r} \right); \label{ergodicb} \\
%\frac{(z +2) h^{(n_{\overline{l}-L-N})}  \log \nu^2  h ^{(n_{\overline{l}-L-N})}}{r\log r}  \leq  \frac{ \frac{z+2}{z}r \log \frac{r\nu^3}{z}}{r\log r} \leq \frac{z+2}{z} \left( 1- \frac{\log z/\nu^3 }{\log r} \right) . \label{ergodica}
\end{eqnarray}
By assumption (\ref{assumptiongeneralxprop}) on $r$, Property $3$, Lemma \ref{productpositive}, balance and choice of $L=l_1\overline{l}$, 
\bes \frac{2  h^{(n_{l-L})}}{r} \leq \frac{2  h^{(n_{l-l_1\overline{l}})}}{h^{(n_{l})}} \leq \frac{2\kappa}{d^{l_1}} < \epsilon.
\ees
Hence the first factors in the RHS of (\ref{ergodicb},\ref{ergodica}) are bounded respectively below by $(1-\epsilon)$ and above by $(1+ \epsilon)$.  
The second factor in the RHS of (\ref{ergodica}) is trivially less than one. 
%Using the upper bound on $z$ in (\ref{upperboundz}), l
%Let us estimate the second factor in (\ref{ergodicb}) using the assumption 
From $ r < h^{(n_{{l}+1})} $,
\be \label{ergodicerror}
 \frac{  \log  h^{(n_{{l}-L})}}{\log r } \geq \frac{ \log  h^{(n_{{l+1}})} - \log \frac{ h^{(n_{{l+1}})}}{ h^{(n_{{l}-L})}}}{\log  h^{(n_{{l}+1})} } .
%\leq  \frac{ d\nu \left( \log \| \ZLp{n_{\overline{l}-L-N}, n_{\overline{l}}} \| + 2 \right) }{\log r}
\ee
%where we used that $ r \leq h^{(n_{{l}+1})} $ by assumption 
Since by the heights transformation formula and Lemma \ref{loglimit} 
\bes
\frac{ \log \frac{ h^{(n_{{l+1}})}}{ h^{(n_{{l}-L})}}}{\log  h^{(n_{{l}+1})} } \leq  \frac{\log \| \ZLp{n_{{l-L}} ,  n_{{l+1}}} \|}{\log  h^{(n_{{l}+1})} }  \xrightarrow{l\rightarrow \infty} 0,
\ees
also the second factor in the RHS of (\ref{ergodica}) is bounded from below by $(1-\epsilon)$ if $l\geq l_o$ for some $l_o\geq l_0$.

%Since $ r >  h^{(n_{\overline{l}-L-N})} > d^{\overline{l}-L-N} $ by positivity (\ref{positive}) and Lemma \ref{productpositive}. Using once more the integrability condition (\ref{integrability}) reasoning in the same way than to estimate (\ref{logsumisepsilon}), we can find $R$ such that if $r>R$, then (\ref{ergodicerror}) is less than $\epsilon$.
%%%%%%%%%%%%%%%%%USE A SEPARATE LEMMA FOR (\ref{logsumisepsilon}),

So far, combining (\ref{BSgenxbb}, \ref{BSgenxba}) with (\ref{ergodicb}, \ref{ergodica}) and (\ref{ergodicerror}) and using the fact that  
%$\log z/\nu^3>0$ for $N$ sufficiently big and
 the resonant term is positive, we proved that
\be \label{ergodicestimate}
(1-\epsilon)^2 r \log r
 \leq { \BS{u}{r}(x)}   \leq  (1+ \epsilon) r\log r  + { \sum_{j=0}^{a^{l-L}+1}  \frac{1}{y_j^{l-L}} }.%{r \log r} .
%(1-\epsilon)^2 
% \leq \frac{ \BS{u}{r}(x)}{r \log r}   \leq  (1+ \epsilon)^2   + \frac{ \sum_{j=0}^{a^{l-L}+1}  \frac{1}{y_j^{l-L}} }{r \log r} .
\ee

\paragraph{Resonant term.}
%: secondary resonances.}
We want to prove the following estimate for the resonant term:
%\begin{Lemma}[Resonant term]
\be \label{estimateresonant} 
0\leq  \sum_{j=0}^{a^{l-L}+1}  \frac{1}{y_j^{l-L}} \leq \epsilon r\log r + \frac{\kappa+1}{x_m}.
\ee
%\end{Lemma}
Let us first group $\{ y_j^{l-L} \}_{j=0,\dots , a^{l_l}+1}$ according to visits to different elements of the partition $\xi_{n_l}(\overline{Z})$. 
%Let us denote visits to $I^{(n_{l})}$, analogously to (\ref{defyil-L},\ref{defy0l-L}), by
%\bes
%y^{l}_i \doteqdot  \Orb{x}{r} \cap  Z_i^{l} \cap I^{(n_{l})}, \quad i=1,\dots  a^{l}; \quad y^{l}_0 \doteqdot (T^{(n_{l})})^{-1}  y^{l}_{1}; \qquad y^{l}_{ a^{l}+1} \doteqdot T^{(n_{l})}  y^{l}_{ a^{l}}.
%y^{l-L}_{i+1} \doteqdot T^{(n_{l-L})}  y^{l-L}_{i}, \quad i=1,\dots  a^{l-L}+1.
%\ees
Since 
\bes
\bigcup_{j=0}^{a^{l-L}+1} y_j^{l-L} \subset \bigcup_{j=0}^{a^{l-L}+1} Z_j^{l-L} \subset \bigcup_{i=0}^{a^{l}+1}  Z_i^{l}. 
\ees
we have the estimate
\bes
\sum_{j=0}^{a^{l-L}+1}  \frac{1}{y_j^{l-L}} \leq \sum_{i=0}^{a^{l}+1} \sum_{{\begin{subarray}{c} y_j^{l-L} \in Z_i^{l} \\ j=0,\dots,a^{l-L}+1  \end{subarray}} }   \frac{1}{y_j^{l-L}}.
\ees
Each of the points $y_j^{l-L} \in Z_i^{l}$ belongs to a different floor of a tower of step $n_l$. Hence, arguing as in the Lemma \ref{estimategaplemma} about the Gap Error, each of the terms $\sum_{y_j^{l-L} \in Z_i^{l} }   \frac{1}{y_j^{l-L}}$ can be bounded from above by applying Lemma \ref{Kocharithmetic} to an auxiliary arithmetic progression with step $1/d \kappa \nu h^{l}_i$. The cardinality of points in each group, by (\ref{entriesBmn}), is bounded by
%\bes
$\# \{ y_j^{l-L}  | y_j^{l-L} \in Z_i^{l} \} \leq \# \{  Z_j^{l-L} \in \xi_{n_{l -L}}( Z_i^{l} ) \} \leq \| \ZLp{n_{l-L}, n_l} \|.
$ %\ees
The initial point $\min \{ y_j^{l-L} | \,  y_j^{l-L} \in Z_i^{l} \}$ is given by the only visit to the base $ Z_i^{l} \cap I^{(n_{l})}$. Denote, as above (see (\ref{defyil-L}, \ref{defy0l-L})),
\bes
y^{l}_i \doteqdot  \Orb{x}{r} \cap  Z_i^{l} \cap I^{(n_{l})}, \quad i=1,\dots , a^{l}; \quad y^{l}_0 \doteqdot (T^{(n_{l})})^{-1}  y^{l}_{1}; \qquad y^{l}_{ a^{l}+1} \doteqdot T^{(n_{l})}  y^{l}_{ a^{l}}.
%y^{l-L}_{i+1} \doteqdot T^{(n_{l-L})}  y^{l-L}_{i}, \quad i=1,\dots  a^{l-L}+1.
\ees
%(the last inequality follows from balance by Lemma (\ref{balancelambdah})). 
%Applying Lemma \ref{Kocharithmetic},  
% \lambda^{(n_{\overline{l}})}/d\nu$ (the last inequality follows from lengths balance (\ref{lengthsbalance})) to which we can apply Lemma \ref{Kocharithmetic}. 
Hence we get
\be \label{resonantgrouping}
\sum_{j=0}^{a^{l-L}+1}  \frac{1}{y_j^{l-L}} \leq  \sum_{i=0}^{a^{l}+1}   
d \kappa \nu h^{l}_i ( \log \| \ZLp{n_{l-L}, n_l} \|+1) +
 \sum_{i=0}^{a^{l}+1} \frac{1}{y_i^{l}}.
% \in I^{(n_l )} \cap  Z_i^{l} }   \frac{1}{y_j^{l-L}}
\ee
%\be \label{resonantvsrlogr1}
% \sum_{j=-1}^{z}  \frac{1}{y_j}   \leq   \frac{1}{\overline{y}} +  \sum_{j=1}^{z+1}  \frac{1}{\overline{y} + j  1/ d\nu^2 h^{(n_{\overline{l}})} } \leq   \frac{1}{\overline{y}} +  
%1 d\nu^2 {h^{(n_{\overline{l}})}} (\log (z+1)  + 1). 
%\ee
Comparing the first term in the RHS of (\ref{resonantgrouping}) to $r\log r$ and recalling (\ref{restimate}) we get
\bes\label{secondaryresonaces}
\frac{ d \kappa \nu \left( \sum_{i=0}^{a^{l}+1}  h^{l}_i \right) ( \log \| \ZLp{n_{l-L}, n_l} \|+1) }{r\log r} \leq d \kappa \nu \left(1+\frac{2h^{(n_l)}}{r}\right) \frac{ \log \| \ZLp{n_{l-L}, n_l} \|+1  }{\log h^{(n_l)}} 
\ees
where $(1+2h^{(n_l)}/r)\leq 3$, so the last term, enlarging $l_o$ if necessary, is less than $\epsilon$ when $l\geq l_o$ by Lemma \ref{loglimit} and Corollary \ref{superexp}. 

The second term in the RHS of (\ref{resonantgrouping}) is bounded in two different ways, according to the ratio between $r$ and $h^{(n_{l+1})}$, using the quantity $\sigma_l$ defined in (\ref{sigmadef}) as a threshold.

%\begin{enumerate}
%\item 
{\it Case 1}. Assume $\sigma_l h^{(n_{{l}+1})} \leq r < h^{(n_{{l}+1})}$. 
%Let us reason exactly as before, this time grouping points according to visits to different elements of $\xi_{l+1}(\overline{Z})$. 
Recalling (\ref{approxl+1}), 
\bes
\{ y_i^{l} |\,\, i=0 , \dots , a^{l}+1  \} \subset \bigcup_{j=0}^{a^{l+1}+1} \{  y_i^{l} |\,\, y_i^{l}  \in  \cap  Z_j^{l+1} , \,  i=0 , \dots , a^{l}+1 \}.
% \{ y_j^{l-L}  \in I^{(n_l)} \} \subset \bigcup_{i=0}^{a^{l+1}+1} \{  y_j^{l-L}  \in I^{(n_l)} \cap  Z_i^{l+1} \}. 
\ees
%The cardinality of $ \{  y_j^{l-L}  \in I^{(n_l)} \cap  Z_i^{l+1} \}$ is bounded by the cardinality of $$$\|\ZLp{n_l,n_{l+1}} \|$
To estimate the contribution from each of the sets in the RHS of (\ref{resonantgrouping}), arguing as above, consider an auxiliary arithmetic progression of step $ d \kappa \nu h^{l+1}_j$. The closest point of each set is given by the visit to $I^{(n_{l+1})}$. Remarking that the number of points in each is bounded by
% at most $\|\ZLp{n_l,n_{l+1}} \|$  points, with an arithmetic progression and keeping aside the closest point of each, given by the visit to $I^{(n_{l+1})}$, 
%\bes
$\# \{ y_i^{l}|\, y_i^{l}  \in   Z_j^{l+1} ,\, i=0,\dots, a^{l}+1   \} \leq \# \xi_{n_l}( Z_j^{l+1} )  \leq \| \ZLp{n_{l}, n_{l+1}} \|,
$ %\ees
we get
\be \label{mainres1} %\begin{split}
 \sum_{i=0}^{a^{l}+1} \frac{1}{y_i^{l}}
% \sum_{y_j^{l-L} \in I^{(n_l} )}   \frac{1}{y_j^{l-L}} %& 
\leq  \sum_{j=0}^{a^{l+1}+1}    
d \kappa \nu h^{l+1}_j ( \log \| \ZLp{n_{l}, n_{l+1}} \|+1)  + 
\sum_{\begin{subarray}{c} y_i^{l} \in I^{(n_{l+1})} \\ i=0,\dots, a^{l}+1 \end{subarray}}  \frac{1}{y_i^{l}}.%\\
%\sum_{y_j^{l-L} \in I^{(n_{l+1})}}  \frac{1}{y_j^{l-L}}.%\\
\ee
For the first term in the RHS of (\ref{mainres1}), by the assumptions on $r$, $ h^{l+1}_j/r \leq  h^{(n_{l+1})}/r \leq 1/\sigma_l$ and $a^{l+1} \leq \kappa $, 
\bes
%\frac{ (a^{l+1}+2) h^{(n_{l+1})} ( \log \| \ZLp{n_{l}, n_{l+1}} \|+1)}
\frac{\sum_{j=0}^{a^{l+1}}    d \kappa \nu h^{l+1}_j ( \log \| \ZLp{n_{l}, n_{l+1}} \|+1)}
{r \log r}\leq \frac{(\kappa+2 ) d \kappa \nu ( \log \| \ZLp{n_{l}, n_{l+1}} \|+1)}{\sigma_l \log h^{(n_{l})}} .%\leq \epsilon
\ees
%Since $\sigma_l \xrightarrow{r \rightarrow \infty} 0$ by Lemma \ref{sigmarlemma} the last term in (\ref{resonantvsrlogr1}) can be made less than $(\epsilon/2) r \log r$ when $r$ is sufficiently large.
The last expression, enlarging again $l_o$ if necessary, is smaller than $\epsilon$ if $l \geq l_o$ by Property $(ii)$ and $(iii)$ in Lemma \ref{sigmarlemma}.

The second term in the LHS of (\ref{mainres1}) can be just estimated with $(\kappa +1)/  x_m$ since, as remarked at the beginning of this proof, $\# \Orb{x}{r} \cap  I^{(n_{l+1})} \leq [\kappa] $. This completes the proof of (\ref{estimateresonant}) in this case.
 
%Remark that since $ \Orb{x}{r} \prec Z^{(n_{\overline{l}})}_{j_0} $, each $y_j$, where $0\leq j \leq z$, belongs to a different floor of the tower $Z^{(n_{\overline{l}})}_{j_0}$. Also $y_{-1}$ belongs to a (different) floor of $Z^{(n_{\overline{l}})}_{j_0}$ since the partition $\xi_{n_{\overline{l}-L-N}} (Z^{(n_{\overline{l}})}_{j_0})$ consists of complete towers pieces. 

%Reasoning as in the Lemma \ref{estimategaplemma} about the Gap Error, $\sum_{j=-1}^{z}  \frac{1}{y_j}$ can be bound above applying Lemma \ref{Kocharithmetic} to an auxiliary arithmetic progression with initial point $\overline{y} \doteqdot \min_j y_j$ and step $\lambda^{(n_{\overline{l}})}_{j_0} \geq 1/ d\nu^2 h^{(n_{\overline{l}})}$ (the last inequality follows from balance by (\ref{lambdacomparedtoh})). 
%Hence we get
%\be \label{resonantvsrlogr1}
% \sum_{j=-1}^{z}  \frac{1}{y_j}   \leq   \frac{1}{\overline{y}} +  \sum_{j=1}^{z+1}  \frac{1}{\overline{y} + j  1/ d\nu^2 h^{(n_{\overline{l}})} } \leq   \frac{1}{\overline{y}} +  
%1 d\nu^2 {h^{(n_{\overline{l}})}} (\log (z+1)  + 1). 
%\ee

%Comparing this bound with $r\log r$, and using respectively the upper bound on $z$ in (\ref{upperboundz}), the assumption $r > \sigma_r h^{(n_{\overline{l}})} $ and the definition of $\sigma_r$ in (\ref{sigmadef}), we get 
%\be 
%\frac{h^{(n_{\overline{l}})} (\log (z+1)  + 1) }{r\log r} \leq  \frac{h^{(n_{\overline{l}})} 3 \log 3 z  }{r\log r} \leq \frac{r}{\sigma_r } \frac { 2 \log  3 \| \ZLp{n_{\overline{l}-L-N}, n_{\overline{l}}} \| }{r \log r } \leq 2 \sigma_r .
%%\xrightarrow{r \rightarrow \infty} 0 ,
%\ee
%Since $\sigma_r \xrightarrow{r \rightarrow \infty} 0$ by Lemma \ref{sigmarlemma} the last term in (\ref{resonantvsrlogr1}) can be made less than $(\epsilon/2) r \log r$ when $r$ is sufficiently large.

%For the second term, 
%Let us consider the term $1/\overline{y}$ in (\ref{resonantvsrlogr1}). If $\overline{y} \neq y_{-1}$, clearly $\overline{y} \notin I^{(n_{\overline{l}})}_{j_0}$, base floor of the tower, hence $\overline{y} \geq \lambda^{(n_{\overline{l}})}_{j_0} \geq 1/ d\nu^2 h^{(n_{\overline{l}})}$, again by (\ref{lambdacomparedtoh}). Using again the assumption $r > \sigma_r h^{(n_{\overline{l}})} $ and the definition of $\sigma_r$,\be \label{ybarnoty-1}
%\frac{1/\overline{y}}{r \log r} \leq \frac{d\nu^2 h^{(n_{\overline{l}})}}{r \log r } \leq d\nu^2 \frac{1}{\sigma_r \log r} \xrightarrow{r \rightarrow \infty} 0,
%\ee
%by Lemma \ref{sigmarlemma}, part $(ii)$.

%In the case $\overline{y} = y_{-1}$, recall that $y_{-1}\doteqdot T^{-i_{-1}}x$, with $i_{-1} \leq h^{(n_{\overline{l} - L -N})}$ by its definition (\ref{i-1def}). Hence, by the assumption $(b)$ of Proposition \ref{BSgeneralxprop}, 
%\be \label{y-1estimate}
%y_{-1} \geq  2 \nu d \sigma_r \lambda_1^{(n_{\overline{l} - L -N})} \geq \sigma_r \lambda_1^{(n_{\overline{l}} )} \geq \sigma_r1/ d\nu^2 h^{(n_{\overline{l}})}
%\ee
%and hence, by Lemma \ref{sigmarlemma}, part $(ii)$, 
%\be \label{ybar=y-1} 
%\frac{1/\overline{y}}{r \log r} \leq \frac{d\nu^2 h^{(n_{\overline{l}})}}{r \log r } \leq d\nu^2 \frac{1}{\sigma_r \log r} \xrightarrow{r \rightarrow \infty} 0 .
%\ee
%The estimates (\ref{ybarnoty-1}, \ref{ybar=y-1}) in the two possible cases shows that also the first term in (\ref{resonantvsrlogr1}) can be made less than $(\epsilon/2) r \log r$ when $r$ is sufficiently large.

%\item 
{\it Case 2}. Assume $ h^{(n_{l})} \leq  r < \sigma_l h^{(n_{l+1})} $. In this case, use the trivial estimate 
%given by $y_j \geq \overline{y}$ and the upper bound (\ref{upperboundz}) for $z$:
\bes \label{trivialresbnd}
 \sum_{i=0}^{a^{l}+1} \frac{1}{y_i^{l}}\leq  (a^{l}+2) \frac{1}{x_m}.
% \sum_{j=0}^{a^{l-L}+1}  \frac{1}{y_j^{l-L}}  \leq  (a^{l-L}+1) \frac{1}{x_m}.
% \sum_{j=-1}^{z}  \frac{1}{y_j}   \leq  (z+2) \frac{1}{\overline{y}}. 
\ees
%We seek a bound below for $x_m$. 
Since $x_m= T^i x$ for some $0\leq i < r$ and in this case $r< \sigma_l h^{(n_{l+1})}$, by the assumption (\ref{assumptiongeneralxprop}) and the definition (\ref{badsetforBSu}) of $\Sigma^+_l$,
\bes
x_m \geq \sigma_l \lambda^{(n_l)} \geq  \sigma_l  \frac{1}{\kappa h^{(n_l)}}, 
\ees
where the last inequality uses the balance of $n_l$ (see Lemma \ref{balancehlambda}).
Moreover, from (\ref{restimate}) and $\kappa$-balance of heights,  
\bes
a^{l}\leq \frac{r}{\min_j h^{(n_{l})}_j} \leq  \frac{\kappa r}{ h^{(n_{l})}}.
\ees
Hence,
\bes
\frac{  \sum_{i=0}^{a^{l}+1} \frac{1}{y_i^{l}}  }{r\log r} \leq \frac{ (\frac{\kappa r}{ h^{(n_{l})}} +2)  \frac{\kappa h^{(n_l)}}{\sigma_l} }{r\log r } \leq \frac{\kappa^2+2\kappa}{\sigma_l \log h^{(n_l)}},
\ees
which, again enlarging $l_o$, is smaller than $\epsilon$ for $l\geq l_o$ by Property $(iii)$ in Lemma \ref{sigmarlemma}.
%\end{enumerate}

In both cases we proved the estimate (\ref{estimateresonant}) for the resonant term. Together with (\ref{ergodicestimate}), for an appropriate choice of $\epsilon$, this completes the proof of Proposition \ref{BSgeneralxprop}.
\end{proofof}

%The following Corollary, obtained modifying slighly the proof above, gives an estimate than (\ref{BSgeneralxestimate}) of Proposition \ref{BSgeneralxprop}, but which holds for all points. It is needed in Section \ref{preliminarypartitionsec}.%, since it can be applyed preliminarly to verify the assumptions of Proposition \ref{BSgeneralxprop}.
\begin{cor}\label{BSr^2}
Let $T\in \M^+$. For each $\varepsilon >0$ there exists $r_0$ such that for all $r\geq r_0$, $x\in I^{(0)}$, 
\be
\BS{u}{r}(x) \leq \varepsilon r^2 + \frac{\kappa +1}{x_m}.
\ee
\end{cor}
The estimate in the Corollary is worst than (\ref{BSgeneralxestimate}) in Proposition \ref{BSgeneralxprop}, but holds for all points and is used in Section \ref{preliminarypartitionsec}.
\begin{proof} Let $h^{(n_l)}\leq r < h^{(n_{l+1})}$
Remark that equations (\ref{ergodicestimate}), (\ref{resonantgrouping}) and (\ref{mainres1}) in the proof of Proposition \ref{BSgeneralxprop} were obtained without using the assumption (\ref{assumptiongeneralxprop}) and hence still hold if $r\geq r_0\doteqdot h^{(n_{l_o})}$. Terms estimated by $r\log r$ are clearly less than $\varepsilon  r^2$ choosing $r_0$ large enough. The second term in the RHS of (\ref{mainres1}) is estimated by $(\kappa+1)/x_m$. Let us estimate the first term in the RHS of (\ref{mainres1}) by
\bes
\frac{h^{(n_{l+1})} \log \| \ZLp{n_{l}, n_{l+1}}  \|}{r^2} \leq \frac{h^{(n_{l+1})} \log \| \ZLp{n_{l}, n_{l+1}} \ \|}{{h^{(n_{l})}}^2} \leq \frac{  \| \ZLp{n_{l}, n_{l+1}}  \| \log \| \ZLp{n_{l}, n_{l+1}} \ \|}{const \, d^l}
\ees 
where we used Corollary \ref{superexp} in the last bound. The limit of this ratio as $l\rightarrow \infty$ is zero by (\ref{integrability}) in Property 4. 
\end{proof}


\subsection{Growth of Birkhoff sums of the derivatives.}\label{BSf'andf''sec}
%From the growth of $\BS{u}{r}$ proved in Proposition \ref{BSgeneralxprop} we get the following.
Let $h_\I^{(n)}$ and $\lambda_\I^{(n)}$ be the sequences of heights and lenghts of towers for $T^{\I}$. For $T^{\I} \in \M^+$, let $\{n'_{l'}\}_{l'\in \mathbb{N}}$ be the sequence of balanced times for $T^{\I}$ given by Proposition \ref{existencebalancedtimes}. Let $\sigma_{l'}=\sigma_{l'}(T^{\I})$. Define
\be\label{DefSigma-}
\Sigma^-_{l'} (T) 
\doteqdot \bigcup_{i=0}^{[\sigma_{l'} h_\I^{(n'_{l'+1})}]} T^{-i} [1- \sigma_{l'} \lambda_\I^{(n'_{l'})},1].
\ee
\begin{cor}[Growth of  $\BS{v}{r}$]\label{BSv} Let $T\in \M^-$. 
For any $\epsilon > 0 $ there exists $l_o' >0$ such that, if $l'\geq l_o'$, for any $r \in \mathbb{N}$ and $x\in I^{(0)}$ such that 
\bes
h_\I^{(n'_{l'})} < r < h_\I^{(n'_{{l'}+1})}  \qquad \mathrm{and} \qquad x \notin  \Sigma^-_{l'}(T) ,
\ees 
denoting by $x_M \doteqdot \max_{0\leq i < r}  T^i x$, we get \footnote{Here $\kappa'$ is the same given by Proposition \ref{existencebalancedtimes}.}
\bes 
(1-\epsilon)r\log r \leq  \BS{v}{r}(x)  \leq 
 (1+\epsilon) r\log r + \frac{\kappa'+1}{1-x_M}.
\ees
%where $\kappa'$ is given by Proposition \ref{existencebalancedtimes}.
\end{cor}
\begin{proof} Corollary \ref{BSv} is simply obtained by restating Proposition \ref{BSgeneralxprop} for $u$ and $T^{\I}$ and using the relation with $v$ given by (\ref{BSuvrel}). Remark that $ \min_i \left( T^{\I} \right)^i(1-x) = \min_i \I T^i (x) = \min_i (1-T^i x) = 1-x_M$
and that $(1-x)\in \Sigma^+_{l'}(T^{\I})$ if and only if $x\in \Sigma^-_{l'}(T)$.
%%\bes \min_i (T^{\I})^i(1-x) = \min_i \I T^i (x) = \min_i (1-T^i x) = 1-x_M.
%\ees
\end{proof}
%Assume that a certain asymmetry $C^+>C^-$ is given. 
%Recall that $C^+\neq C^-$ are the asymmetry constants of the roof function (see Section \ref{1asymmetriclog}).
%%%%ELIMINABILE
We can combine the estimates for $\BS{u}{r}$ and $\BS{v}{r}$ as follows.
\begin{cor}[Growth $\BS{f'}{r}$]\label{BSf'cor}
For each $T\in \M$ and $C^+\neq C^-$, there exists $C_1,C'_1 , C_2,C_3 >0$ and $r_o$ such that for $r\geq r_o$, if 
\be\label{assumptionsBSf'}
\begin{array}{ccc}
\begin{array}{rcl}
 h^{(n_l)}&\leq r < &h^{(n_{l+1})}, \\
%\qquad 
 h_\I^{(n'_{l'})}&\leq r < &h_\I^{(n'_{{l'}+1})} 
%\qquad \mathrm{and} \qquad 
\end{array} & \mathrm{and} &
x \notin  \Sigma^+_{l}(T) \cup \Sigma^-_{l'}(T)  ,
\end{array}
\ee 
%denoting by $x_m \doteqdot \min_{0\leq i < r} T^i x$, 
%there exists $C>0$ such that 
and $x$ is not a singularity of $\BS{f}{r}$,\footnote{Here $x_m$, $x_M$, $\kappa$ and $\kappa'$ are as in Corollary \ref{BSv}.}
%\be \label{BSf'}
\begin{eqnarray}\label{BSf'}
\BS{f'}{r}(x)  &\leq& 
 -C_1 r\log r + \frac{C^+(\kappa'+1)}{1-x_M}, \qquad \mathrm{if} \,\, C^+>C^-;
\\
 \BS{f'}{r}(x)  &\geq& 
 C_1 r\log r - \frac{C^-(\kappa+1)}{x_m}, \qquad \mathrm{if} \, \, C^+ < C^- ; 
\\
| \BS{f'}{r}(x) | &\leq& \label{BSf'upperbound}
 C_2 r\log r +\frac{C_2 (\kappa+1)}{x_m} + \frac{C_2 (\kappa'+1)}{1-x_M} .% \qquad \mathrm{if} \,\, C^+>C^-;
%\ee
\end{eqnarray}
%where $x_m$, $x_M$, $\kappa$ and $\kappa'$ are as in Corollary \ref{BSv}. 
Moreover, for all $x\in I^{(0)}$ different from singularities of $\BS{f}{r}$,
\be \label{roughboundf'}
| \BS{f'}{r}(x)|  \leq 
 C_3 r^2 +  \frac{C_3( \kappa+1)}{x_m} + \frac{C_3 (\kappa'+1)}{1-x_M}.
\ee
\end{cor}
%\begin{rem}
%An analogous Corollary can be stated if $C^+<C^-$. In this case (\ref{BSf'}) has to be substituted by
%\bes
% \BS{f'}{r}(x)  \geq 
% C r\log r - \frac{\kappa}{x_m} .
%\ees
%\end{rem}
\begin{proof}
Assume $C^+>C^-$. Consider the sequence  $\alpha_r \rightarrow 0$ in Proposition \ref{reducetouv}. 
%Since $r(x,t)\geq t/3$ by Property 4 in Proposition \ref{preliminarypartitionprop} and $C^+>C^-$, o
One can choose $r_1$ so that, for each $r \geq r_1$, we have $(C^+ - \alpha_r ) > (C^- + \alpha_r)$.
%Choose $t \gg 1$ so that $\alpha_r \rightarrow 0$ $(C^+ - \alpha_r ) > (C^- + \alpha_r)$, which is possible since by Proposition \ref{reducetouv}, $\alpha_r \rightarrow 0$ as $r \rightarrow \infty$ and $r(x,t)\geq t/3$ by Property 4 in Proposition \ref{preliminarypartitionprop}.%Choose $\nu >1$ such that  $(C^+  - \alpha_r )/\nu > \nu ( C^- + \alpha_r) $. 
Hence it is also possible to choose $\epsilon >0$ so that, for $r \geq r_1$, 
%the ratio 
$C_1 \doteqdot (C^+ - \alpha_r )(1-\epsilon)  - ( C^- + \alpha_r ) (1+ \epsilon)  > 0$.  
%%%\bes
%C_1 \doteqdot (C^+ - \alpha_r )(1-\epsilon)  - ( C^- + \alpha_r ) (1+ \epsilon)  > 0 .
%\ees
%$(C^+ - \alpha_r )(1-\epsilon)^2/( C^- + \alpha_r ) (1+ \epsilon)^2 \nu^2> 1$
%\bes
%\frac{(C^+ - \alpha_r )(1-\epsilon)^2}{( C^- + \alpha_r ) (1+ \epsilon)^2 \nu^2} > 1.
%\ees
%Let $r_o\doteqdot \max \{ r_o, h^{(n_{l_o})}(T),  h^{(n'_{l'_o})}(T^{\I}) \}$. 
By Proposition \ref{reducetouv}, Proposition \ref{BSgeneralxprop} and Corollary \ref{BSv}, which can be applied by the assumptions (\ref{assumptionsBSf'}) when $r\geq r_o$, $r_o\doteqdot \max \{ r_1, h^{(n_{l_o})},  h_\I^{(n'_{l'_o})} \}$, 
%\doteqdot \max \{ r_o, h^{(n_{l_o})}(T),  h^{(n'_{l'_o})}(T^{\I}) \}$  
\bes
\begin{split}
&\BS{f'}{r}(x)  \leq -( C^+\!-\alpha_r )  \BS{u}{r}(x) + (  C^-\!+\alpha_r ) \BS{v}{r}(x) \leq  -( C^+\!-\alpha_r ) {(1-\epsilon)} r \log r + \\& + (  C^-\!+\alpha_r ) (1+ \epsilon)   r \log r + \frac{(C^-\!+\alpha_r) (\kappa'+1)}{1-x_M}  \leq - C_1  r \log r + \frac{C^+  (\kappa'+1)}{1-x_M} .
\end{split}
\ees
The case $C^+<C^-$ can be treated analogously.

Also (\ref{BSf'upperbound}) follows similarly: enlarging $r_1$ so that if $r\geq r_1$, $\alpha_r \leq \min (C^+, C^- )$, by Proposition \ref{reducetouv},
\be\label{bounduv}
| \BS{f'}{r}(x) | \leq ( C^+ + C^- ) (  \BS{u}{r}(x) + \BS{v}{r}(x) )
\ee
and $\BS{u}{r}$, $\BS{v}{r}$ can be again estimated by Proposition \ref{BSgeneralxprop} and Corollary \ref{BSv}.

For (\ref{roughboundf'}), apply to (\ref{bounduv}) the rough estimate on $\BS{u}{x}$ for all points given by Corollary \ref{BSr^2}  and the analogous one for $\BS{v}{x}$ which follows from $T\in \M^-$.

\end{proof}
%%%%The following estimate for the Birkhoff sums of the second derivative follows trivially from Corollary \ref{BSf'} and is enough in Section \ref{partitionsec}.
% from the estimates for the first derivative and are enough in Section \ref{partitionsec}.
\begin{cor}[Growth $\BS{f''}{r}$]\label{growthBSf''}
For each $T\in \M$ and $C^+\neq C^-$, there exists $C_4 >0$ and $r_o$ such that for $r\geq r_o$, if (\ref{assumptionsBSf'}) holds,
%\be\label{assumptionsBSf'}
%\begin{array}{ccc}
%\begin{array}{c}
% h^{(n_l)}(T)\leq r < h^{(n_{l+1})}(T), \\
%\qquad %
% h^{(n'_{l'})}(T^{\I})\leq r < h^{(n'_{{l'}+1})}(T^{\I}) 
%Under the same assumptions than in Corollary \ref{BSf'cor}, there exists also $C_4>0$ such that 
\bes
| \BS{f''}{r}(x) | \leq C_4
%r\log r
 \max \left\{\frac{1}{x_m}, \frac{1}{1-x_M} \right\} \left( r\log r + \frac{\kappa+1}{x_m} + \frac{\kappa'+1}{1-x_M} \right).
\ees
\end{cor}
\begin{proof}
By definition of logarithmic singularity, there exists $\delta>0$ such that $0\leq f''(x)\leq 2C^+/x^2$ if $x<\delta$ and $0\leq f''(x)\leq 2C^-/(1-x)^2$ if $x >1-\delta$. Let $M_{f''}$ be the maximum of $|f''|$ on $[\delta, 1-\delta]$. Hence, for each $x$ not a singularity of $\BS{f}{r}$,
\be \label{trivialbndf''}
\begin{split}
|\BS{f''}{r}(x)| & \leq 2C^+ \BS{{1}/{x^2}}{r}(x) +  2C^- \BS{{1}/{(1-x)^2}}{r}(x) + r M_{f''} \leq \\
& \leq  2C^+ \frac{1}{x_m} \BS{u}{r}(x) +  2C^-  \frac{1}{1-x_M} \BS{v}{r}(x)  + r M_{f''}   .
\end{split}
\ee 
Applying Proposition \ref{BSgeneralxprop} and Corollary \ref{BSv} one get the desired estimate.  
%Since trivially $\BS{{1}/{x^2}}{r}(x) \leq 1/x_m \BS{r}{u}$ and $\BS{{1}/{(1-x)^2}}{r}(x) \leq 1/(1-x_M) \BS{r}{v}$, the Corollary follows from {trivialbndf''}
\end{proof}

%%%%%%

\section{Construction of the mixing partitions.}\label{mixingpartitionsec}
%%%%%%%ELIMINABILE
In this Section we construct the partitions $\eta_{m}(t)$ which verify the mixing criterium (see Lemma \ref{mixingcriterium}). %which satisfy (\ref{mainestimate}),
The construction is carried out in three main steps, formulated in Section \ref{propertiessec} as Propositions \ref{preliminarypartitionprop}, \ref{growthBSf'partition} and \ref{mainpropertiespartitionprop}. Their proofs are in Sections \ref{preliminarypartitionsec}, \ref{stretchingpartitionsec} and \ref{mainpropertiespartitionsec} respectively. We anticipate in Section \ref{areaestimatesec} the final area estimates, which conclude the proof of Theorem \ref{mixing}.

%In this Section we construct the partial partitions $\eta_{m}(t)$ of $I^{(0)}$ into intervals which satisfy (\ref{mainestimate}), so that mixing for the special flow follows from the mixing criterium stated as Lemma \ref{mixingcriterium}.

%The construction of the partitions is carried out through further refinements, in three main steps. The properties of the partitions obtained in each step are preliminary stated in Propositions \ref{preliminarypartitionprop}, \ref{growthBSf'partition} and \ref{mainpropertiespartitionprop} in Section \ref{propertiesandareasec}. 

%To motivate the estimates in Proposition \ref{mainpropertiespartitionprop}, we anticipate in Section \ref{propertiesandareasec} the final area estimates, which conclude the proof of the Main Proposition \ref{mixing} by making quantitative the intuitive explanation of the mixing mechanism in Section \ref{mixingcriteriumsec}.
% are presented in Section \ref{areaestimates}, to motivate the need of the estimates in Proposition \ref{mainpropertiespartition}.

%The construction of the partitions is postponed to Sections \ref{preliminarypartitionsec}, \ref{stretchingpartitionsec} and \ref{mainpropertiespartitionsec}, which contain the proofs of Propositions \ref{preliminarypartitionprop}, \ref{growthBSf'partition} and \ref{mainpropertiespartitionprop} respectively.
% and carryed out in various steps. Section \ref{preliminarypartitionsec} containes the proof of Proposition \ref{preliminarypartitionprop}, in Section \ref{stretchingpartitionsec} Proposition \ref{growthBSf'partition}, while Proposition \ref{mainpropertiespartitionprop} is proven troughout Section \ref{mainpropertiespartitionsec}.

%\subsubsection{Step 0: Partition into continuity intervals of controlled size.} \label{continuitysec}
%\paragraph{Upper bound on discrete iterations}
%Consider a fixed continuous time $t$. The maximum number of discrete iterations of $T$ when flowing by $t$, i.e.
%\bes
%r_M(t) \doteqdot \sup_{x\in I^{(0)}} r(x,t), 
%\ees
%can be bounded above in the following way. By using that $f\geq m_f >0$ and the definition of $r(x,t)$, for each $x$
%\bes r(x,t) m_f \leq \BS{f}{r(x,t)} (x) \leq t .
%\ees
%Hence, defining
%\bes
%R_M(t) \doteqdot \left[ \frac{t}{m_f} \right] + 1,
%\ees
%where $[\cdot]$ denotes the integer part, $r_M(t)\leq R_M(t)$.

%\paragraph{Continuity intervals}
%It is easy to see that any iterate $T^n\doteqdot T\circ \dots \circ T$ is again an IET: denoting by $\beta_0 =0 < \beta_1 < \dots < \beta_{d-1} <\beta_d=1$ the discontinuities of $T$, the discontinuities of $T^N$ are 
%\be \label{discontinuities}
%\{ T^{-j} \beta_i |\quad i=0, \dots d; \quad  0\leq j\leq N \}.
%\ee%
%In particular $T^N$ is an exchange of at most $N d+1$ intervals.%

%Let $\eta_0(t)$ be the \emph{partition} of $I^{(0)}$ \emph{into continuity intervals} for $T^{R_M(t)}$, i.e. the partition into semi-open intervals whose endpoints coincide with the set (\ref{discontinuities}) where $N=R_M(t)$. By construction, for each $0\leq j\leq R_M(t)$, $T^j$ restricted to any $[a,b[ \in \eta_0(t)$ is continuous.

%\paragraph{Control of the interval size}
%In order to control from below the size of continuity intervals, we will extract from $\eta_0$ a partial partition, throwing away intervals which are too small.

%Let $h(t)$ be any positive function of $t$ such that $h(t)\rightarrow +\infty$ as $t \rightarrow +\infty$. Further restrictions on $h$ are put in \ref{hrestrictions}.
%\begin{lemma}\label{sizecontrol}
%\end{lemma}

%%%%%%%ELIMINABILE
%\subsection{Partitions properties and area estimates.}\label{propertiesandareasec}
%We write that $\eta' \subset \eta$ and $\eta '$ is a \emph{refinement} of the partial partitions into intervals $\eta$ if for each $I\in \eta'$, we have $I \in \eta$, i.e. $\eta'$ is obtained from $\eta$ by throwing away some intervals. 
\subsection{Partitions properties.}\label{propertiessec}
%\paragraph{Upper bound on discrete iterations.}
%Consider a fixed continuous time $t$. The maximum number of discrete iterations of $T$ when flowing by $t$, i.e. $r_M(t) \doteqdot \sup_{x\in I^{(0)}} r(x,t)$, can be bounded above for each $x$ by using that $f\geq m_f >0$ and the definition of $r(x,t)$:
%\bes r(x,t) m_f \leq \BS{f}{r(x,t)} (x) \leq t .
%\ees
%Hence, defining
%\be \label{RMdef}
%R_M(t) \doteqdot \left[ \frac{t}{m_f} \right] + 2,
%\ee
%where $[\cdot]$ denotes the integer part, $r_M(t)+1\leq R_M(t)$.
%%%%ELIMINABILE
As a preliminary step, we construct in Section \ref{preliminarypartitionsec} partial partitions which satisfy the following proposition.
Denoting $[\cdot]$ the integer part, let 
\be \label{RMdef}
R_M(t) \doteqdot \left[ t/m_f
%\frac{t}{m_f}
 \right] + 2.
\ee
%where $[\cdot]$ denotes the integer part,
\begin{prop}[Preliminary partitions.]\label{preliminarypartitionprop}
For each $0<\delta<1$ and $M>1$, there exist $t_0>0$ and partial partitions $\eta_p(t)$ for $t \geq t_0$, such that $Leb(\eta_p(t)) > 1-\delta$ 
%, the mesh, i.e. $\sup_{I\in \eta_p(t)} Leb(I)$, converges to zero as $t\rightarrow + \infty$ 
and the following properties hold for each $I=[a,b[ \, \in \eta_p(t)$.
\begin{enumerate}
%%%ELIMINABILE: TITOLETTI
\item \emph{Continuity intervals:}

$T^j$ is continuous on $[a,b]$ for each $0\leq j \leq  R_M(t)$; 
\item \emph{Control of interval sizes:}  
\bes \frac{1}{t \log\log t} \leq Leb(I) \leq \frac{2}{t \log\log t };
\ees
\item \emph{Control of the distance from singularities:}
\bes dist (T^j I, 0 ) \geq \frac{M}{t \log \log t}, \quad dist (T^j I, 1 ) \geq \frac{M}{t \log \log t}, \quad  0\leq j \leq  R_M(t);
\ees
\item \emph{Control of the number of discrete iterations:} 
\bes
\frac{t}{3} \leq r(x,t) \leq R_M(t) \leq \frac{2}{{m_f}} t, \qquad \forall x \in I  .
\ees
\end{enumerate}
\end{prop}
%%%%%%%%%Senza titoli
%\begin{enumerate}
%\item
%$T^j$ is continuous on $I$ for each $0\leq j \leq  R_M(t)$; (Partitions into continuity intervals)
%\item $\frac{1}{2t \log\log t} \leq Leb(I) \leq \frac{2}{t \log\log t }$ (Control of interval sizes);
%\item $dist (T^j I, 0 ) \leq \frac{1}{t \log t}$ and $dist (T^j I, 1 ) \leq \frac{1}{t \log t}$ for all $j \leq  R_M(t)$. (Control of the distance from singularities).
%\item $\frac{t}{3} \leq r(x,t) \leq \frac{2}{{m_f}} t$ (Control of the number of discrete iterations) .
%\end{enumerate}
%\end{prop}

%%%%%%%%%%%%%%OldProp
%As a preliminary step, we construct in Section \ref{preliminarypartition} partial partitions which satisfies the following proposition.
%\begin{prop}[Preliminary partition.]
%For each $t \gg 1$ there esists a partial partition $\eta_t^1$ of the unit interval such that $\eta_t^1$ converge to the trivial partition into points as $t$ tends to infinity and the following properties hold for each $I=[a,b[ \in \eta_t^1$
%\be \label{controlintervalsizes}
%\frac{1}{2t \log\log t} \leq Leb([a,b[) \leq \frac{2}{t \log\log t }; \qquad \mathrm{(Control \, of \, interval \, sizes)},
%\ee
%\be
%dist (T^j I, 0 ) \leq \frac{1}{t \log t} , \quad dist (T^j I, 1 ) \leq \frac{1}{t \log t}, \quad \forall j \leq  \max_{x\in I} r(x,t), 
%\ee
%and $T^j$ is continuous on $I$ for each $j \leq  \max_{x\in I} r(x,t)$.
%\end{prop}

%\paragraph{Uniformely Stretching partition.}
%Assume now that a certain asymmetry $C^+>C^-$ is given and a
Assume now that $T\in \M$. % (see Definition \ref{Mdef}).  
For definiteness, assume also that the asymmetry constants of the roof function satisfy $C^+>C^-$.  
%%%% ELIMINABILE
%The case $C^+<C^-$ can be treated analogously. 
%Using the estimates on the growth of Birkhoff sums obtained in Section \ref{growthBSsec}, we can refine the partitions $\eta_p(t)$ to get the following.
%%%ELIMINABILE
%We write that $\eta' \subset \eta$ and $\eta '$ is a \emph{refinement} of the partial partitions into intervals $\eta$ if for each $I\in \eta'$, we have $I \in \eta$, i.e. $\eta'$ is obtained from $\eta$ by throwing away some intervals. 
Using the estimates on the growth of Birkhoff sums obtained in Section \ref{growthBSsec}, we can refine the partitions $\eta_p(t)$ to get the following.
\begin{prop}[Stretching partitions.]\label{growthBSf'partition}
For $T\in \M$ and $C^+> C^-$, there exists $C_1', C_2', C'' >0$ such that for each $0<\delta<1$, $M>1$, if $\eta_p(t)$ are the corresponding partitions in Proposition \ref{preliminarypartitionprop}, 
%For each $T\in \M$, for each $\delta>0$, 
% with full measure such that when considering the suspension flow with roof $u$ and $v$ respectively and base $T$ in this set, 
there exists a sequence of refined partitions $\eta_{s}(t) \subset \eta_p(t)$ with $Leb(\eta_{s}) > Leb(\eta_{p}) -\delta$ and there exists $t_1>t_0$ such that, when $t\geq t_1$, for any $x \in \eta_{s}(t)$ and 
%for which there exists $C_1>0$ 
 integer $r$
%\in \mathbb{N}$, 
 with  $t/3  \leq r \leq 2t / m_f $,% and for each $x \in \eta_{s}(t)$, 
\begin{eqnarray} \label{f'decreases}
% \BS{f'}{r}(x)  \geq \frac{C^- -C^+}{2} r \log r; \qquad 
\BS{f'}{r}(x)  &\leq &- C_1'\,
%\left(\frac{C^+ -C^-}{2} \right) 
r \log r ;\\
| \BS{f'}{r}(x) |  &\leq & C_2'\,
%\left(\frac{C^+ -C^-}{2} \right) 
r \log r   \label{SBf'boundabove}; \\
%\ee 
%and%\footnote{Recalling that $M$ is given by Proposition \ref{preliminarypartitionprop}.}
%\be 
\label{BSf''}
\BS{f''}{r}(x) & \leq &\frac{ C''}{M} r^2 (\log r) ( \log \log r ).
\end{eqnarray}
\end{prop}
%%%
Let us show that Proposition \ref{growthBSf'partition} implies in particular that $r(\cdot,t)$ is an increasing function on each interval $I\in \eta_{s}(t)$ for $t\geq t_1$. Assume $x< y$ are points of $I$. 
Since in particular $\BS{f'}{r(x,t)}<0$, the function $\BS{f}{r(x,t)}$ is strictly decreasing. Hence
%By definition of $r(\cdot,t)$, $\BS{f}{r(x,t)}(x)>t$.
%Since in particular $\BS{f'}{r(x,t)}(x)>0$, $\BS{f}{r(x,t)}$ is an increasing function. Hence
%\be 
$\BS{f}{r(x,t)}(y) <  \BS{f}{r(x,t)}(x) \leq t $. 
%\ee
Using again the definition of $r(\cdot,t)$, we get $r(y,t)\geq r(x,t)$.

%In particular, we justified (\ref{rarbmaxmin}) in Section \ref{geometricdescriptionsec}.
%Recalling also the notation (\ref{Deltafdef},\ref{Deltafjdef}) introduced in the same Section,

\paragraph{Geometric description of the dynamics of partition elements.} \label{geometricdescriptionsec}
Let $I=[a,b[$ be an element of the partition $\eta_{s}(t)$.
% that we constructed in the previous sections.
Consider $\varphi_t I$ and let us give first a geometric description of $\varphi_t I$ for $t \gg 1$. 
For $C^+ > C^-$, 
%Assume that $C^+ > C^-$. 
as just proved, 
%, when $C^+ > C^-$ and $t\gg 1$,
%we prove in Section \ref{stretchingpartitionsec}
 $r(\cdot,t)$ is an increasing function on each $[a,b[\in \eta_p(t)$. Hence, let\footnote{The dependence on $t$ will be omitted when $t$ is clear from the context.}
\begin{eqnarray} \label{rarbmaxmin}
&&r(a)= r(a,t)= \inf_{x\in [a,b[} r(x,t); \qquad r(b) = r(b,t)=\sup_{x\in [a,b[} r(x,t);
\\ \nonumber
&&J=J([a,b[,t) \doteqdot r(b,t) -r(a,t) +1 .
\end{eqnarray}
The image $\varphi_t I$ splits into several curves and $J$ gives exactly their cardinality. More precisely, consider the equation $\BS{f}{r}=t$ on $I$, which has a solution exactly for $r=r(a)+1, \dots, r(b)$, unique by monotonicity. Denote by $y_j$ the solution of
\be \label{splittingpointsdef}
\BS{f}{r(a)+j}(y_j)=t , \qquad j = 1 , \dots, J-1, 
\ee
% if we denote by $y_j$ a point such that $\BS{f}{r(y_j,t)} = t$, 
%\bes 
%\BS{f}{r(y_j,t)} = t
%\ees
so that by (\ref{flowdef}), $r(y_j,t)=r(a)+j$ and 
%\bes
$
\varphi_t (y_j, 0 ) =( T^{r(y_j,t)}(y_j) , 0).
$ %\ees
%It is easy to see that 
The points $a\doteqdot y_0 \leq y_1 \leq  \dots \leq y_j \leq y_{j+1} \leq y_{J-1} \leq y_{J} \doteqdot b $
%there are exactly $J$ solutions, $y_1, \dots, y_J$ and moreover $a=y_0 \leq \dots \leq y_j \leq y_{j+1} \leq \dots \leq b $. Also, it is easy to see that
%$t= \BS{f}{r(a)+j-1} (y_j)$, 
%\bes \label{splittingpointdef} t= \BS{f}{r(a)+j-1} (y_j), 
%\ees
%hence $r(y_j,t)=r(a)+j$. 
%The points $y_1, \dots, y_{J-1}$ 
 are splitting points, meaning that the image $\varphi_t I$ consists of $J$ curves, which are the graphs of $t - \BS{f}{r(a)+j}$ restricted to $I_j \doteqdot [y_{j}, y_{j+1}[$, $j=0,\dots, J-1$. Each curve projects to $T^{r(a)+j} (I_j)$, by (\ref{flowdef}). In particular, %since $t - \BS{f}{r(a)+j}|_{I_j}$ projects to $T^{r(a)+j} (I_j)$,
 the curves project to the orbit $T^{r(a)+j} (I)$, for $j=0,\dots, J\!-\!1$. 

%%%%%%ELIMINABILE
%Recall that we want to show that the proportion of $\varphi_t I$ which falls inside a rectangle $R$ is proportional to the area of $R$, i.e. to prove (\ref{mainestimate}).
%Since the curves projects along an orbit, showing first that the number of curves $J(I,t)$ increases to infinity, by unique ergodicity of the IET we will be able to show that they distribute uniformly when projected to the base. On the other side, showing that each of these curves has a slope tending to infinity and they tend to be better and better approximated by a vertical line, will allow to show also uniform distribution in the vertical direction. 

\paragraph{Properties of the mixing partitions.}
Given $\delta>0$, if $[b_1,b_2]$ is the base of the rectangle $R$, denote $ \chi$ the indicator of $[b_1+\delta,b_2-\delta]$. Choose $t_2 \geq t_1$ so that $2/(t_2 \log \log t_2) < \delta$ and the mesh of the partitions $\eta_{p}(t)$ for $t\geq t_2$ is bounded by $\delta$ by Property 2 in Proposition \ref{preliminarypartitionprop}. Denoting by $h(R)$ the height of the rectangle $R$, for $j=0,\dots, J\!-\!1$, let
\be \label{Ijhdef}
I_{j}^{h(R)} \doteqdot \{ x | \quad t - h(R)  \leq \BS{f}{r(a)+j}(x) \leq t  \}.
%I_{j}^{h(R)} \doteqdot \{ x | \quad \BS{f}{r(a)+j-1}(y_j) - h(R)  \leq \BS{f}{r(a)+j-1}(x) \leq \BS{f}{r(a)+j-1} (y_j) \}.
\ee
%It is easy to see using the flow definition (\ref{flowdef}) and recalling that $h(R)\leq m_f$, that p
Points in $I_{j}^{h(R)}$ are the ones that reach the correct height to intersect $R$, i.e. if $x\in I_{j}^{h(R)}$, then $\varphi_t(x,0)$ is contained in the horizontal strip $I^{(0)}\times h(R)$ (as shown in the proof of Lemma \ref{areaboundlemma}).
%%%%%%ELIMINABILE SPAZIO SOPRA
%\begin{lemma}\label{areaboundlemma}
%For each $I=[a,b[ \in \chi_t$,
%\be\label{areabound}
%\mu ([a,b[ \cap \varphi^{-t} R) \geq 
%\sum_{j=0}^{J([a,b[)} \chi_{[b_1,b_2-\delta]}
%\left( T^{r(a)+j} \right) Leb \left( I_j ^{h(R)} \right) . 
%\ee
%\end{lemma}
%
%Denote by $\BS{f}{r(a)+j}(b^-) = \lim_{b\rightarrow b^-}  \BS{f}{r(a)+j}(b)$, which is well defined by Properties $1$ and $3$ in Proposition \ref{preliminarypartitionprop}.
For $j=0,\dots, J-1$, %let $\BS{f}{r(a)+j}(b) = \lim_{b\rightarrow b}  \BS{f}{r(a)+j}(b)$, which is well defined by Properties $1$ and $3$ in Proposition \ref{preliminarypartitionprop}. 
denote by 
%\begin{eqnarray}
%\Delta f = \Delta f([a,b[,t) & \doteqdot & \BS{f}{r(a,t)}(a) -  \BS{f}{r(a,t)}(b) ;  \label{Deltafdef} \\
\bes
\Delta f^j = \Delta f^j([a,b[,t) \doteqdot \BS{f}{r(a)+j}(a) -  \BS{f}{r(a)+j}(b) ; \label{Deltafjdef} \qquad \Delta f %= \Delta f([a,b[,t) 
\doteqdot \Delta f^0. \ees
%\end{eqnarray}
%Both $\Delta f$ and $\Delta f^j$ depends on $t$ and on $[a,b[$, but the dependence is not written explicitely.  
Remark that $\Delta f^j\geq 0$. The quantity $\Delta f$ express the the delay accumulated between the endpoints in time $t$. %: the point $T^{r(a)}(b)$ still has to travel along the fibers when, after $r(a)$ discrete iterations, the endpoint $a$ stops. 
%In other words, %it gives the vertical stretch of the curve $t- \BS{f}{r(a)-1}|_{[a,b[}$. 
% accumulated because of the that the point $T^{r(a)}(b)$ still has to travel along the fibers when, after $r(a)$ discrete iterations, the endpoint $a$ stops. In other words, it expresses the delay accumulated because of the strech of Birkhoff sums. 
%Similarly, 
Also, the quantity $\Delta f^j$ gives the vertical stretch of the graph of $t - \BS{f}{r(a)+j}|_{[a,b[}$.
% after $r(a)$ discrete iterations. The other endpoint, after $r(a)$ discrete iterations, reaches $T^{r(a)}(b)$ and has to cover ,  has  
%The quantity $\Delta f$ express the lenght that the point $T^{r(a)}(b)$ still has to travel along the fibers

The last refinement of the partitions is needed to guarantee the following properties.
%%%%ELIMINABILE prima e TITOLETTi
\begin{prop}[Mixing partitions]\label{mainpropertiespartitionprop}
Let $T\in \M^+$, $C^+>C^-$. Given $\epsilon>0$ and $0<\delta<1$, there exists $M(\epsilon)$, $t_3>t_2$ and refined partial partitions $\eta_m (t) \subset \eta_{s}(t)$, where $\eta_{s}(t)$ are the partitions given by Proposition \ref{growthBSf'partition}, such that $Leb(\eta_m(t))> Leb(\eta_{s}(t))-2\delta$ and for $t\geq t_3$ and each $I=[a,b[ \, \in \eta_m(t)$, $J(I,t)\rightarrow + \infty$ as $t\rightarrow +\infty$ and the following properties hold: %and $j=1,\dots, J(I,t)-2$, 
\begin{enumerate}

\item \emph{Uniform vertical distribution}: %for $j=1,\dots, J(I,t)\!-\!2$,
\be\label{uniformstretch}
\left|\frac{ h(R)(b-a) }{\Delta f^j(I,t)} - Leb(I^{h(R)}_j) \right| \leq \frac{ h(R)(b-a) }{\Delta f^j(I,t)} \epsilon , \qquad j=1,\dots, J(I,t)-2; %\qquad \mathrm{( Uniform \, stretch)}. 
\ee

\item \emph{Variation of slopes}: 
\be \label{variationslopes}
\left| \frac{\Delta f(I,t)}{\Delta f^j(I,t)} -1 \right|     \leq  \epsilon, \quad j=0,\dots, J(I,t)\!-\!2; % \qquad \mathrm{( Variation \, of \, slope )} ;
\ee

\item \emph{Asymptotic number of curves}:
\be \label{asymptoticscurves}
\left| \frac {J([a,b[,t)-1}{\Delta f(I,t)} -1 \right| \leq \epsilon ; %\qquad \mathrm{( Asymptotic \, number \, of \, curves)} ;
\ee

\item \emph{Equidistribution on the base}: for some $\overline{x}\in I$,
\be \label{equidistributionbase}
\left| \frac{1}{J([a,b[,t)-1} \left( \sum_{j=0}^{J([a,b[,t)-2} \chi
%_{[b_1-\delta,b_2-\delta]}
 \left( T^{r(a)+j}(\overline{x}) \right)\right) - (b_2-b_1 - 2\delta) \right|   \leq    \epsilon . %\qquad \mathrm{( Equidistribution \, on \, the \, base )}
\ee
\end{enumerate}
%Moreover, $J(I,t)\rightarrow + \infty$ as $t\rightarrow +\infty$.
%\be \label{equidistributionbase}
%\left| \frac{1}{J([a,b[,t)} \sum_{j=0}^{J([a,b[,t)} \chi_{[b_1,b_2-\delta]} \left( T^{r(a)+j}(b) \right) - (b_2-b_1 - \delta) \right|   \leq    \epsilon ; \qquad \mathrm{( Equidistribution \, on \, the \, base )}
%\ee
%\be \label{variationslopes}
%\left| \frac{\Delta f}{\Delta f^j} -1 \right|     \leq  \epsilon, \quad j=1,\dots, J; \qquad \mathrm{( Variation \, of \, slope )} ;
%\ee
%\be \label{asymptoticscurves}
%\left| 1- \frac {J([a,b[,t)}{\Delta f} \right| \leq \epsilon ; \qquad \mathrm{( Asymptotic \, number \, of \, curves)} ;
%\ee
%\be\label{uniformstretch}
%\left|\frac{ h(R)(b-a) }{\Delta f^j} - Leb(I^{h(R)}_j) \right| \leq \frac{ h(R)(b-a) }{\Delta f^j} \epsilon \qquad \mathrm{( Uniform \, stretch)}. 
%\ee
\end{prop}

\subsection{Area estimates.}\label{areaestimatesec}
%%%%%ELIMINABILE
%Let us conclude the proof of Theorem \ref{mixing} from Proposition \ref{mainpropertiespartitionprop}.% the proof of Theorem \ref{mixing}. 
Let us show that the properties in Proposition \ref{mainpropertiespartitionprop} are enough to deduce the estimate (\ref{mainestimate}) of the Mixing Criterium (Lemma \ref{mixingcriterium}) and hence conclude the proof of the Theorem \ref{mixing}. 
\begin{lemma}\label{areaboundlemma}
For each $I=[a,b[ \in \eta_{m}(t)$, $t\geq t_3$ and $x\in I$,
\be\label{areabound}
Leb ([a,b[ \cap \varphi^{-t} R) \geq 
\sum_{j=1}^{J([a,b[,t)-2} \chi%_{[b_1,b_2-\delta]}
\left( T^{r(a)+j}x \right) Leb \left( I_j ^{h(R)} \right) . 
\ee
\end{lemma}
\begin{proof}
Assume that $j_0$ is such that $1\leq j_0 \leq J([a,b[,t)-2$ and $\chi (T^{r(a)+j_0}x) =1 $, i.e. $T^{r(a)+j_0}x \in [b_1+\delta,b_2-\delta]$. Recalling that $t_2$ was chosen so that $\sup_{I\in \eta_{m}(t)} Leb(I) < \delta$ for $t\geq t_2$, when  $t\geq t_3 \geq t_2$, we have $T^{r(a)+j_0} I \subset [b_1,b_2]$. 

It is enough to show that $I_{j_0} ^{h(R)} \subset [a,b[ \cap \varphi^{-t} R$ to conclude. 
If $y\in I_{j_0} ^{h(R)}$, by definition $t - h(R)  \leq \BS{f}{r(a)+j_0}(y) \leq t$. Also, recalling that $h(R)< m_f$,
%%%ELIMINABILE
\bes
t \leq t - h(R) + f(T^{r(a)+j_0}y)  < \BS{f}{r(a)+j_0+1}(y) ,
\ees
which shows that $r(y,t)=r(a)+j_0$ and also that $y\in [y_{j_0} ,  y_{j_0+1}] \subset [a,b[$, by monotonicity %of $r(\cdot,t)$ 
and definition of splitting points (\ref{splittingpointsdef}). 
%by monononicity of $\BS{f}{r(a)+j_0 -1}$, $x_j\leq y \leq x_{j+1}$ and in particular $y\in  [a,b[$ and $r(y,t)=r(a)+j_0$. 

It follows, by definition (\ref{Ijhdef}) of $I_{j_0} ^{h(R)}$ and (\ref{flowdef}) of the flow action, that
%%%%ELIMINABILE
\bes
\varphi_t(y,0) = (T^{r(a)+j_0}y , t-   \BS{f}{r(a)+j_0 }(y)   ) \in [b_1,b_2] \times [0,h(R)]= R.
\ees
%Moreover, by monononicity of $\BS{f}{r(a)+j_0 -1}$, $x_j\leq y \leq x_{j+1}$.
This shows that $I_{j_0} ^{h(R)} \subset [a,b[ \cap \varphi^{-t} R$.
%, which is enough to conclude. 
\end{proof}
Let us estimate the RHS of (\ref{areabound}). For $t\geq t_3$,
\begin{eqnarray}
&& \sum_{j=1}^{J([a,b[)-2} \chi%_{[b_1,b_2-\delta]}
\left( T^{r(a)+j}(x) \right) Leb \left( I_j ^{h(R)} \right) \geq \nonumber \\
%%%%%%%%Togli pure questa!
%& \geq & (1-\epsilon) \sum_{j=1}^{J([a,b[)-2} \chi_{[b_1,b_2-\delta]}
%\left( T^{r(a)+j}(x) \right) \frac{ h(R) (b-a)}{\Delta f^j} \geq \label{useuniformstretch} \\
&\geq  & (1-\epsilon) h(R) (b-a) \sum_{j=1}^{J([a,b[,t)-2} \chi%_{[b_1,b_2-\delta]}
\left( T^{r(a)+j}(x) \right) 
%\frac{ \Delta f}{\Delta f} 
\frac{1}{\Delta f^j} \geq   \label{useuniformstretch} \\
&\geq & (1-\epsilon)^2 h(R) (b-a) \sum_{j=1}^{J([a,b[)-2} \chi%_{[b_1,b_2-\delta]}
\left( T^{r(a)+j}(x) \right) 
%\frac{ J(I)}{\Delta f} 
\frac{1}{\Delta f}
 \geq   \label{usevariationslopes}
 \\
&\geq & (1-\epsilon)^3 h(R) (b-a) \sum_{j=1}^{J([a,b[)-2} \frac{ \chi%_{[b_1,b_2-\delta]}
\left( T^{r(a)+j}(x) \right) }{J([a,b[,t)-1}  %\chi_{[b_1,b_2-\delta]}\left( T^{r(a)+j}(x) \right)  
\geq  \label{useasymptoticscurves} \\
&\geq & (1-\epsilon)^3 h(R) (b-a) (b_2-b_1 - 2\delta - 2\epsilon ) \xrightarrow{\epsilon , \delta \rightarrow 0} \mu(R) (b-a)  \label{useequidistributionbase} .
\end{eqnarray}
We used, in order, Property $1$ %(uniform vertical stretch) 
%(\ref{uniformstretch}) 
to get (\ref{useuniformstretch}), 
Property $2$ %(control of the variation of slopes (\ref{variationslopes}) 
to get (\ref{usevariationslopes}),
% the asymptotic number of curves (\ref{asymptoticscurves}) 
Property $3$ to get (\ref{useasymptoticscurves}) and, eventually, to get (\ref{useequidistributionbase}) we combined 
Property $4$ %the equidistribution on the base (\ref{equidistributionbase}) 
with $\chi ( T^{r(a)}x )/(J(I,t)-1)\leq \epsilon$ for $t\geq t_3$ if $t_3$ is enlarged if necessary, since $J(I,t)$ tends to infinity.% to get (\ref{useequidistributionbase}).

%%%
When $\epsilon$ and $\delta$ are chosen sufficiently small, 
% the very last expression can be made close to $\mu(R) (b-a)$.
together with the Lemma \ref{areaboundlemma}, this concludes the proof of (\ref{mainestimate}). From Lemma \ref{mixingcriterium}, we get Theorem \ref{mixing}.

\subsection{Preliminary partitions.}\label{preliminarypartitionsec}
% (Proof of Proposition \ref{preliminarypartitionprop}).}\label{preliminarypartitionsec}
Let us prove Proposition \ref{preliminarypartitionprop}.  

%constructing the preliminary partitions. 
%\begin{proofof}{Proposition}{preliminarypartitionprop}
%\paragraph{Continuity intervals of controlled size.
%} \label{continuitysec}
Consider a fixed continuous time $t$. The maximum number of discrete iterations of $T$ when flowing by $t$, i.e. $r_M(t) \doteqdot \sup_{x\in I^{(0)}} r(x,t)$, can be bounded from above for each $x$ by using that $f\geq m_f >0$ and the definition of $r(x,t)$. We get 
\bes r(x,t) m_f \leq \BS{f}{r(x,t)} (x) \leq t.
\ees
Recalling the definition (\ref{RMdef}), we get 
%where $[\cdot]$ denotes the integer part, 
$r_M(t)+1\leq R_M(t)$.
%Consider a fixed continuous time $t$. The maximum number of discrete iterations of $T$ when flowing by $t$, i.e.
%\bes
%r_M(t) \doteqdot \sup_{x\in I^{(0)}} r(x,t), 
%\ees
%can be bounded above in the following way. By using that $f\geq m_f >0$ and the definition of $r(x,t)$, for each $x$
%\bes r(x,t) m_f \leq \BS{f}{r(x,t)} (x) \leq t .
%\ees
%Hence, defining
%\be \label{RMdef}
%R_M(t) \doteqdot \left[ \frac{t}{m_f} \right] + 2,
%\ee
%where $[\cdot]$ denotes the integer part, $r_M(t)+1\leq R_M(t)$.

\paragraph{Continuity intervals of controlled size.} \label{continuitysec}
%\paragraph{Continuity intervals.}
It is easy to see that any iterate $T^n\ \doteqdot T \cdot \phantom{,}\! \dots \phantom{,}\! \cdot T$ is again an IET: denoting by $\beta_0 =0 < \beta_1 < \! \phantom{,}\dots \!\phantom{,} < \beta_{d-1} <1$ the discontinuities of $T$, the discontinuities of $T^N$ are 
\be \label{discontinuities}
\{ T^{-j} \beta_i |\quad i=0, \dots , d-1; \quad  0\leq j < N \}.
\ee
Remark that $T^N$ is an exchange of at most $N d +1$ intervals.

%%%Shorten continuity intervals
Let $\eta_0(t)$ be the \emph{partition} of $I^{(0)}$ \emph{into continuity intervals} for $T^{R_M(t)}$, i.e. the partition into semi-open intervals whose endpoints coincide with the set (\ref{discontinuities}) where $N=R_M(t)$. By construction, for each $0\leq j\leq R_M(t)$, $T^j$ restricted to any $[a,b[ \, \in \eta_0(t)$ is continuous.%, i.e. Property 1 of Proposition \ref{preliminarypartitionprop} holds. 

%%%%SPAZIO
%\paragraph{Control of the interval size.}
%%%%%In order to control from below the size of continuity intervals, we will refine $\eta_0(t)$, throwing away intervals which are too small. 
Given $M>1$, consider the following set
\bes
U_1 \doteqdot \bigcup _{\begin{subarray}{c} 0\leq i \leq d \\ 0\leq j \leq R_M(t) \end{subarray}} \overline{Ball}(T^{-j}\beta_i , \frac{2M}{t \log \log t} ),
\ees
which consists of closed balls of radius $2M/t \log \log t$ centered at the endpoints of $\eta_0(t)$.
%Let $h(t)$ be any positive function of $t$ such that $h(t)\rightarrow +\infty$ as $t \rightarrow +\infty$. Further restrictions on $h$ are put in \ref{hrestrictions}.
%%%%
Let $\eta_1(t)$ be the partial partition obtained from $\eta_0(t)$ by throwing away all intervals completely contained in $U_1$. Since, using (\ref{RMdef}),
%%%ELIMINABILE
\be \label{measureS1tendstozero}
Leb(U_1) \leq \frac{4 M}{t \log \log t} d \left( \frac{t}{m_f} + 3 \right) \xrightarrow{t \rightarrow + \infty} 0,
\ee
it follows that 
%, given $\delta>0$, one can choose $t_0$ such that 
$Leb(\eta_1(t))\geq  1- Leb(U_1)$ %\geq 1-\delta/2$ for $t\geq t_0$.% 
converges to one.
 % as $t$ tends to infinity. 
Moreover, by construction, each $I\in \eta_1(t)$ contains at least one $y\notin U_1$. Hence, since the endpoints of $I$ are centers of the balls in $U_1$, $Leb(I) \geq 4M/t \log \log t$.
%some $T^{-{j_1}}\beta_{i_1}$ and $T^{-{j_2}}\beta_{i_2}$ with $0\leq i_1, i_2 \leq d$ and $0\leq j_1, j_2 \leq R_M(t)$ by definition of $\eta_0(t)$, 
%%%ELIMINATO \bes \label{lowerboundIlenght}
%Leb(I) = dist (y, T^{-{j_1}} \beta_{i_1}) +  dist (y, T^{-{j_2}} \beta_{i_2}) > \frac{4 M}{t \log \log t}.
%\ees

%%%ELIMINABILE
\paragraph{Distance from singularities.}
%We want now to eliminate points which, flowing for time $t$, gets too close (closer than $t\log t$) to the singularities of the roof function. 
Let 
\bes
U_2 \doteqdot \bigcup _{0\leq j \leq R_M(t)}
%\begin{array}{c} 1\leq i \leq d \\ 0\leq j \leq R_M(t) \end{array}}
T^{-j}\left[0 , \frac{M}{t \log \log t}\right)  \cup 
  \bigcup _{0\leq j \leq R_M(t)}  T^{-j}\left[1- \frac{M}{t \log \log t},1 \right).
\ees
%where $Ball^+(x,r)$ and $Ball^-(x,r)$ denote respectively the half balls $[x,x+r)$ and $(x-r,x]$. 
Let $\eta_2(t) = \eta_1(t) \backslash U_2$. By construction, if $x\in \eta_2(t)$, 
\bes dist (T^{s}x, 0 ) \geq \frac{M}{t \log \log t}, \quad  dist (T^{s}x, 1 ) \geq \frac{M}{t \log \log t}, \qquad 0\leq  s\leq R_M(t),
\ees
which is Property $2$ of Proposition \ref{preliminarypartitionprop}.
%Let us verify that $\eta_2(t) \xrightarrow{t \rightarrow + \infty} \eta$, trivial partition into points. 
Similarly to (\ref{measureS1tendstozero}), also $Leb(U_2)\xrightarrow{t\rightarrow \infty} 0$. %, as $t$ tends to infinity.%, as in (\ref{measureS1tendstozero}).
%Since $U_2 \subset U_1$, 
%\bes
%Leb(U_2 ) \leq  \frac{1}{t \log \log t}  \left( \frac{t}{m_f} + 3 \right) \xrightarrow{t \rightarrow + \infty} 0,
%\ees
Given $\delta >0$, choose $t_0$ so that $Leb(\eta_2(t))\geq Leb(\eta_1(t))- Leb(U_2)> 1-\delta/2$ for $t\geq t_0$. 
%it follows from (\ref{measureS1tendstozero}) that $Leb(\eta_2(t)) \geq Leb(\eta_1(t))- Leb(U_2)$ tends to one. 
%Given $\delta >0$, choose $t_0$ so that $Leb(\eta_2(t))>1-\delta/2$ for $t\geq t_0$. 
%%%
%Since moreover $M/ t \log \log t \geq 2M/ t \log \log t$, 
%Remark that, if $t\gg 1$ so that $\log \log t \geq 2 \log \log t$, 
%Moreover the i
%Since each $I\in \eta_1(t) $ is a continuity interval for $T^{j}$, $0\leq j \leq R_M$, 
%does not contain in its interior any point of the form $T^{-j}\beta_i$, $0\leq i\leq d-1$, $0\leq j \leq R_M$, 
Intervals $I \in \eta_2(t)$ are either intervals of $\eta_1(t)$ or are obtained by some $I'\in \eta_1(t)$ by cutting an interval of length at most $M/(t \log \log t)$ on one or both sides of $I'$. 
%%%SALVA SPIEGAZIONE
%To see it, since $I'=[a',b'[\in \eta_1(t) $ is a continuity interval for $T^{j}$, $0\leq j \leq R_M$, $T^j I'$ does not contain any discontinuity point, in particular $0$. Hence $T^j a'>0$ and since $Leb(I')> 4M/t \log \log t$, the part eliminated cannot be a central interval.  
Hence, %by (\ref{lowerboundIlenght}),
$Leb(I') \geq 2M/t \log \log t$.
%%%ELIMINATO\bes
%Leb(I) \geq Leb (I') - \frac{2M}{t \log \log t} > \frac{2M}{t \log \log t}.
%\frac{1}{t \log \log t} \left( 1- \frac{2\log \log t}{ \log \log t} \right)  \xrightarrow{t \rightarrow + \infty} \frac{1}{t \log \log t}.
%\ees 
%Let $t\gg 1$ so that $Leb(I) \geq 1/(2t \log \log t)$.

Let $\widetilde{\eta_2}(t)$ be the union of intervals of the form $[a,b'[\subset [a,b[$ associated to each $[a,b[\in \eta_2(t)$. Choosing each $b'$ close enough to each $b$, one still has $Leb([a,b'[)> 2M/t \log \log t$ and $Leb(\widetilde{\eta_2}(t)) > 1-\delta/2$. Since $T^j$, for $0\leq j \leq  R_M(t)$, is continuous on $[a,b']$, Property $1$ holds for $[a,b'[\, \in \widetilde{\eta_2}(t)$.%is satisfacted.

Construct $\eta_3(t)$ from $\widetilde{\eta}_2(t)$ by cutting each of the intervals $I \in \widetilde{\eta}_2(t)$ in pieces which satisfy the Lengths Control Property $2$ of Proposition \ref{preliminarypartitionprop}. % of lenght at most $2/t\log \log t$. 
For example, cut first $[Leb(I)/ (1/t\log \log t)] -1 $ intervals of length exactly $1/(t\log \log t)$ starting from the left, so that the last remaining interval has length at most $2/t\log \log t$.

%Since each $I\in \eta_3(t)$ is contained in some $I_i$, $I_i\in \eta_i(t)$ for $i=0,2$, 
Properties $1$ and $3$ still hold and 
%\be\label{Lebeta3}
$Leb(\eta_3(t)) = Leb(\widetilde{\eta}_2(t)) >1-\delta/2$ for $t\geq t_0$.%  \quad t\geq t_0$.
%\ee
%Hence to conclude the proof of Proposition \ref{preliminarypartitionprop} we now want to get Property $4$.
%and moreover $Leb(\eta_3(t)) = Leb(\eta_2(t))>1-\delta$. To conclude the proof of Proposition \ref{preliminarypartitionprop}. 

%Let us resume the properties obtained so far in the following.
%\begin{lemma}\label{preliminary1}
%For each $I \in \eta_3(t)$ ($t \gg 1$),
%\begin{enumerate}
%\item
%$T^j$ is continuous on $I$ for each $0\leq j \leq  R_M(t)$,
%\item $\frac{1}{2t \log\log t} \leq Leb(I) \leq \frac{2}{t \log\log t }$;
%\item $dist (T^j I, 0 ) \leq \frac{1}{t \log t}$ and $dist (T^j I, 1 ) \leq \frac{1}{t \log t}$ for all $j \leq  R_M(t)$.
%\end{enumerate}
%Moreover, $\eta_3(t) \xrightarrow{t \rightarrow + \infty} \eta$, trivial partition into points.
%\end{lemma}

\paragraph{Control of the number of discrete iterations.}
Let us bound from below $r(x,t)$ when $x\in \eta_3(t)$. As a consequence of Property $3$,
%Let us show that when $x\in \eta_3(t)$ it is possible to give a lower bound on $r(x,t)$. As a consequence of Property $3$,% of Lemma \ref{preliminary1},  
\be \label{controlf}
f(T^j x) \leq const \log (t \log \log t) , \qquad 0\leq j \leq R_M(t). 
\ee
Hence $\BS{f}{r(x,t)+1}(x) \leq (r(x,t)+1)  const \log ( t \log \log t)$ and since by definition of $r(x,t)$, we have $\BS{f}{r(x,t)+1}(x)>t$, 
\be \label{roughboundbelowforr}
r(x,t) \geq \frac{t}{const \log (t \log \log t)} -1 \xrightarrow{t \rightarrow + \infty} + \infty,
\ee
uniformly for all $x\in \eta_3(t)$.
%%%%\paragraph{Control of one point in each $I$.}
Since $f\in L^1$ and $T$ is ergodic, by Birkhoff ergodic theorem, for each $\delta >0$ there exists a measurable set $E_{\delta}$ and $N_{\delta}>0$ such that $Leb (E_{\delta})< \delta/2$ and 
%for each $x \notin E_{\delta}$ and $r\geq N_{\delta}$,
%%%ELIMINABILE
\bes
\left| \frac{1}{r} \BS{f}{r}(x) - \int f(s) \ud s \right|  < 1, \qquad \forall \, x \notin E_{\delta}, \quad r\geq N_{\delta}.
\ees
%Again by Lemma \ref{preliminary1}, choose $t\gg 1$ so that there esists $I_1, \dots, I_s \in \eta_3(t)$ such that 
%$Leb (E_{\delta}) \Delta \bigcup_{i=i}^{s} I_i) < \delta$.
%\bes
%Leb (E_{\delta}) \Delta \bigcup_{i=i}^{s} I_i) < \delta .
%\ees
Define a refined partial partition
%%%\bes
$ \eta_4(t) \doteqdot  \eta_3(t) \backslash \{ I  \in \eta_3(t) | \quad I \subset  E_{\delta} \}.$
%\ees
For $t\geq t_0$, %together with (\ref{Lebeta3}),
 $Leb( \eta_4(t) ) \geq Leb(\eta_3(t))-\delta/2\geq  1 - \delta$. By construction for each $I \in \eta_4(t)$ there is at least one $x_I \in I$ such that $| \frac{1}{r} \BS{f}{r}(x_I) - 1| < 1$ for all $r\geq N_{\delta }$.
Enlarging $t_0$ if necessary, by (\ref{roughboundbelowforr}) we can assure $r(x_I,t) > N_{\delta}$ for each $x_I$, $I \in \eta_4(t)$. Hence %$|\BS{f}{r(x_I,t)+1}(x_I)-(r(x_I,t)+1)|< (r(x_I,t)+1)$
$\BS{f}{r(x_I,t)+1}(x_I)< 2(r(x_I,t)+1)$, which, together with $\BS{f}{r(x_I,t)+1}(x_I) >t $ gives
\be \label{rxI}
 r(x_I,t) > t/2-1 , \quad \forall x_I , \, I \in \eta_4(t).
\ee 

%%%\paragraph{Control of the variation for other points.}
To control all other $r(x,t)$, $x \in \eta_4(t)$, let us estimate the variation $r(x,t)-r(x_I,t)$ when $x \in I\in \eta_4(t)$. 
Assume $r(x,t) < r(x_I,t)$, otherwise we already have the lower bound.
%%%
By Properties 1 and 3, %in Lemma \ref{preliminary1}
 $\BS{f}{r}$ is continuous on $I$ and $(T^r)'=1$ for $0\leq r\leq R_M$, so $(\BS{f}{r}(x))'= \BS{f'}{r}(x)$. By mean value theorem there exists $z$ between $x_I$ and $x$ such that
%%%ELIMINABILE
\bes
\left| \BS{f}{r(x,t)}(x_I) - \BS{f}{r(x,t)}(x) \right| \leq 
%\sum_{i=0}^{r(x,t)-1} f'(T^i z) 
| \BS{f'}{r(x,t) } (z)  |
|x-x_I|.
\ees
%Enlarging again $t_0$ by (\ref{roughboundbelowforr}), we can apply 
Apply the rough bound on $\BS{f'}{r}$ in Corollary \ref{BSr^2}, enlarging again $t_0$ by (\ref{roughboundbelowforr}) so that $r(x,t)\geq r_0$ for $t\geq t_0$. Combining it with Property $3$ already proved, which gives $1/x_m \leq t \log\log t /M$ and $1/(1-x_M) \leq t \log\log t /M$, we get that $|\BS{f'}{r(x,t)}(z)| \leq const \, t^2$ for $t\geq t_0$. Since $Leb(I)\leq 2/( t \log \log t)$,
\be \label{roughboundvariation}
\left| \BS{f}{r(x,t)}(x_I) - \BS{f}{r(x,t)}(x) \right| \leq \frac{const \, t}{\log \log t}.
\ee
Hence, using (\ref{roughboundvariation}) and $\BS{f}{r(x_I,t)}(x_I)\leq t$ 
%to get (\ref{variationsecondline})
 and then $ \BS{f}{r(x,t)}(x)> t - f(T^{r(x,t)} x ) $ and (\ref{controlf}),% to get (\ref{variationthirdline}),
\bes
\begin{split}
&(r(x_I,t) - r(x,t)) m_f  \leq  \BS{f}{r(x_I,t)}(x_I) -  \BS{f}{r(x,t)}(x_I)  \leq \\
&\, t - \BS{f}{r(x,t)}(x) + \frac{const \, t}{\log \log t} \leq 
%  t-t+ 
const \log(t \log t \log t ) + \frac{const \, t}{\log \log t}   =  o(t).
\end{split}
\ees
%%%%ELIMINATO Sostituisci con quello sopra, ma elimina le referenza
%\begin{eqnarray}
%(r(x_I,t) - r(x,t)) m_f  &\leq & \BS{f}{r(x_I,t)}(x_I) -  \BS{f}{r(x,t)}(x_I) \nonumber \leq \\ &\leq& 
%t - \BS{f}{r(x,t)}(x) + \frac{const \, t}{\log \log t}     \label{variationsecondline} < \\&<& 
%  t - t + const \log(t \log t \log t ) + \frac{const \, t}{\log \log t}   =  o(t). \label{variationthirdline} 
%\end{eqnarray}
%%%ELIMINABILE
Rearranging and using the control for $x_I$ given by (\ref{rxI}), $r(x,t) \geq r(x_I,t) -o(t) \geq t/2 -1 - o(t)$. 
%%%%
Hence, recalling also $r(x,t)\leq R_M(t)\leq  t/m_f +2$, if $t_0$ is large enough
%we get Property $4$.
, when $t\geq t_0$,  for each $x \in I \in \eta_4(t)$,
$t/3 \leq r(x,t) \leq R_M(t) \leq 2t/m_f$,
%%%\bes
%\frac{t}{3} \leq r(x,t) \leq \frac{2}{{m_f}} t  ,
%\ees 
which is Property $4$. Since 
% For each $I \in \eta_4(t)\subset \eta_i(t)$, $i=0,1,2,3$, 
also the other Properties still hold, setting $\eta_p(t)\doteqdot \eta_4(t)$ this proves Proposition \ref{preliminarypartitionprop}.


%\begin{lemma}\label{preliminary2}
%For each $\delta>0$, there exists $\eta_4(t)\subset \eta_3(t)$ such that $Leb(\eta_4(t)) > 1-\delta $ and for each $t \gg 1 $, for each $x \in I \in \eta_4(t)$,
%\bes
%\frac{t}{3} \leq r(x,t) \leq \frac{2}{{m_f}} t  .
%\ees 
%\end{lemma}

%Lemma \ref{preliminary1} and Lemma \ref{preliminary2} together, by setting $\eta_p(t)= \eta_4(t)$, prove Proposition \ref{preliminarypartitionprop}.

\subsection{Stretching partitions.}\label{stretchingpartitionsec}
% (Proof of Proposition \ref{growthBSf'partition}).}\label{stretchingpartitionsec}
Let us prove Proposition \ref{growthBSf'partition}. 

%We first prove two Lemmas.
Let $T\in \M$. For each $t$, let $l(t)$ and $l'(t)$ be uniquely determined by
\be \label{tbetweenh}
h^{(n_{l(t)})} \leq  R_M(t) < h^{(n_{l(t) + 1})}; \qquad h_\I^{(n'_{l'(t)})} \leq  R_M(t) < h_\I^{(n'_{l'(t) + 1})},
\ee
where  $\{n_l\}_{l\in \mathbb{N}}$ and $\{n'_{l'}\}_{l'\in \mathbb{N}}$ are the sequences of balanced times given by Proposition \ref{existencebalancedtimes} for $T$ and $T^{\I}$ respectively.
\begin{lemma}\label{controlhbetweenrlemma}
There exists $L\in \mathbb{N}$ independent of $t$ such that if $\frac{t}{3}\leq r \leq R_M(t)$ then
\be\label{controlhbetweenr}
h^{(n_{l(t)-L})} \leq  r < h^{(n_{l(t) + 1})}; \qquad h_\I^{(n'_{l'(t)-L})} \leq  r < h_\I^{(n'_{l'(t) + 1})}
\ee
\end{lemma}
\begin{proof} %The upper estimate is trivial. 
Let $l\in \mathbb{N}$ be such that $d^{l}\geq \max \{ 6\kappa /m_f, 6\kappa' /m_f \} $\footnote{Recall that $\kappa$ and $\kappa'$are given by Proposition \ref{existencebalancedtimes} and Corollary \ref{BSv}.}. 
By Property (\ref{positive}) in Proposition \ref{existencebalancedtimes}, we can apply Lemma \ref{productpositive} considering products of positive matrices that appear every $\overline{l}$ balanced steps and get, recalling the choice of $l$, by balance of the induction steps and (\ref{tbetweenh}),
$h^{(n_{l(t)-l\overline{l}})}$%\leq  {\kappa}\min_j h_j^{(n_{l(t)-l\overline{l}})} 
$\leq (\kappa/d^l) \min_j h_j^{(n_{l(t)})} \leq  (\kappa/d^l)(2t/m_f) \leq t/3 \leq r$.
Analogous expressions can be obtained also for $h_\I^{(n'_{l'(t)-l\overline{l}})}$ and show that setting $L\doteqdot l\overline{l}+1$ we get (\ref{controlhbetweenr}).
\end{proof}
%Old Longer version
%\be \label{tbetweenh}
%h^{(n_{l(t)})} \leq  t < h^{(n_{l(t) + 1})}; \qquad h^{(n'_{l'(t)})} \leq  t < h^{(n'_{l'(t) + 1})},
%\ee
%where  $\{n_l\}_{l\in \mathbb{N}}$ and $\{n'_{l'}\}_{l'\in \mathbb{N}}$ are the sequences of balanced times given by Proposition \ref{existencebalancedtimes} for $T$ and $T^{\I}$ respectively.
%\begin{lemma}\label{controlhbetweenrlemma}
%There exists $L\in \mathbb{N}$ independent of $t$ such that if $\frac{t}{3}\leq r \leq \frac{2}{m_f}t$ then
%\be\label{controlhbetweenr}
%h^{(n_{l(t)-L})} \leq  r < h^{(n_{l(t) + L})}; \qquad h^{(n'_{l'(t)-L})} \leq  r < h^{(n'_{l'(t) + L})}
%\ee
%\end{lemma}
%\begin{proof}
%Let $l\in \mathbb{N}$ be such that $d^{l}/\kappa > 2/m_f$, $\kappa/d^{l}<1/3$, $d^{l}/\kappa' > 2/m_f$ and $\kappa'/d^{l}<1/3$\footnote{Recall that $\kappa$ and $\kappa'$are given by Proposition \ref{existencebalancedtimes} and Corollary \ref{BSv}.}. 
%By Property (\ref{positive}) in Proposition \ref{existencebalancedtimes}, we can apply Lemma \ref{productpositive} considering products of positive matrices that appear every $\overline{l}$ balanced steps and get, recalling the choice of $l$, by balance of the induction steps and (\ref{tbetweenh}),
%$h_j^{(n_{l(t)+l\overline{l}+1})}  \geq  \min_j h_j^{(n_{l(t)+l\overline{l}+1})} \geq d^l \min_j h^{(n_{l(t)+1})} \geq \frac{ d^l}{\kappa}  h^{(n_{l(t)+1})} \geq \frac{2 t}{m_f} \geq r$. Similarly, $h_j^{(n_{l(t)-l\overline{l}})}\leq  {\kappa}\min_j h_j^{(n_{l(t)-l\overline{l}})} \leq \frac{\kappa}{d^l} \min_j h^{(n_{l(t)})} \leq \frac{t}{3} \leq r$.
%Analogous expressions can be obtained also for $h_j^{(n'_{l'(t)+l\overline{l}+1})}$ and $h_j^{(n'_{l'(t)-l\overline{l}})}$ and show that setting $L\doteqdot l\overline{l}+1$ we get (\ref{controlhbetweenr}).
%\end{proof}
%Since moreover $r\leq R_M(t) \doteqdot [t/m_f]+2$, if $l(R_M)$ and $l'(R_M)$ are uniquely defined by 
%\bes 
%h^{(n_{l(R_M)})} \leq  R_M(t) \leq h^{(n_{l(R_M) + 1})}; \qquad h^{(n'_{l'(R_M)})} \leq  R_M(t) \leq h^{(n'_{l'(R_M) + 1})},
%\ees
%we can improve the Lemma as follows.
%\begin{cor}\label{refinedhbetweenr} For each $t/3  \leq r \leq 2 t / m_f $,%
%\bes
%h^{(n_{l(t)-L})} \leq  r < h^{(n_{l(R_M) + 1})}; \qquad h^{(n'_{l'(t)-L})} \leq  r < h^{(n'_{l'(R_M) + 1})}
%\ees
%where $l(R_M) \leq l(t)+L$ and $l'(R_M)\leq l'(t)+L$.
%\end{cor}

Define the set  $\Sigma_t = \Sigma_t (T)$ as 
%Recall that the set $\Sigma_l(T)$ was defined in (\ref{badsetforBSu}). % in Section \ref{BSgeneralxpropsec}.
% Define $\Sigma_t = \Sigma_t (T)$ as
\be\label{defSt}
\Sigma_t \doteqdot \bigcup_{l=l(t)-L}^{l(t)-1} \Sigma_l^+(T) \cup \bigcup_{l=l'(t)-L}^{l'(t)-1} \Sigma^-_{l'}(T) \cup  \overline{\Sigma}^+_{l(t)}(T) \cup \overline{\Sigma}^-_{l'(t)}(T),
\ee
where the sets $\Sigma^+_l(T)$ and  $\Sigma^-_{l'}(T)$ were defined in (\ref{badsetforBSu}) and (\ref{DefSigma-}) and where
\bes
 \overline{\Sigma}^+_{l(t)}(T) \doteqdot  \bigcup_{i=0}^{\min\{ R_M(t), [\sigma_{l(t)} h^{(n_{l(t)+1})}]\}} T^{-i} [0, \sigma_{l(t)} \lambda^{(n_{l(t)})}].
\ees
and $\overline{\Sigma}^-_{l'(t)}$ is the analogous truncation of $\Sigma^-_{l'(t)}$. Remark that
%We define these last two sets differently to assure that the cardinality of intervals in each $\Sigma_l$ is at most $R_M$. Indeed we get the following.
%\begin{lemma}\label{Sigmatlemma}
%The set $\Sigma_t$ is a union of at most $(4L+2)R_M$ intervals. Also,
\be\label{LebSigmat}
Leb(\Sigma_t) \xrightarrow{t\rightarrow +\infty} 0,
\ee
%\end{lemma}
%\begin{proof}
%Each $\Sigma_l$, for $l\leq l(R_M)-1$ is a union of at most $h^{(n_{l+1})}\leq h^{(n_{l(R_M)})} \leq  R_M$ intervals. Also $\overline{\Sigma}_{l(R_M)}$ is by definition a union of at most $R_M$ intervals. Similarly for $\Sigma_{l'}$ and $\overline{\Sigma}_{l'(R_M)}$. The set $\Sigma_t$ is union of at most $2(2L+1)$ such sets, by its definition and Corollary \ref{refinedhbetweenr}. 
%To estimate its measure, simply 
because $Leb(\Sigma^+_l)\xrightarrow{t\rightarrow +\infty} 0$ for each $l\geq l(t)-L$ from (\ref{LebSigma}), the same holds for each $Leb(\Sigma^-_{l'})$, $l'\geq l'(t)-L$ and $\Sigma_t$ is union of at most $2(L+1)$ such sets. 
%\end{proof}
\begin{proofof}{Proposition}{growthBSf'partition}
Fix $T$, $C^+>C^-$, $\delta>0$ and $M>1$ and let $\eta_p(t)$ be the preliminary partitions given by Proposition \ref{preliminarypartitionprop} for $t\geq t_0$. %, consider the preliminary partitions $\eta_p(t)$ given by Proposition \ref{preliminarypartitionprop} and 
Consider the set $\Sigma_t (T)$ defined in (\ref{defSt}) and, by (\ref{LebSigmat}), choose $t_1\geq t_0$ so that $Leb(\Sigma_t)<\delta/2$ for $t\geq t_1$.  %, by Lemma \ref{Sigmatlemma}, choose $t_1\geq t_0$ so that the set $\Sigma_t (T)$ defined in (\ref{defSt}). 
% and cover it with intervals $I \in \eta_p(t) $. 
Define $\eta_{s}(t)$ as the partition obtained from $\eta_p(t)$ throwing away all the intervals which intersect $\Sigma_t$. %Choosing $t\geq t_1$ for some $t_1>0$, we ensure $Leb(\Sigma_t)<\delta/2$  by Lemma \ref{Sigmatlemma}. 
If $I\in \eta_p(t)$ and $I\cap \Sigma_t\neq \emptyset$, then, from Property $1$ in Proposition \ref{preliminarypartitionprop}, either $I\subset \Sigma_t$ or, for some $0\leq j\leq R_M(t)$, $T^j I$ contains either some points $\sigma_{l} \lambda^{(n_{l})}$ with $l(t)-L\leq l \leq l(t)$ or some $1-\sigma_{l'} \lambda_{\I}^{(n_{l'})}$ with $l'(t)-L\leq l' \leq l'(t)$. Hence, using (\ref{LebSigmat}), Property $2$ in Proposition \ref{preliminarypartitionprop} and bounding the number of such points, we get 
%intervals which cover $\Sigma_t$ are either completely contained in it or intersect one of the endpoints of $\Sigma_t$. Hence, using Lemma \ref{Sigmatlemma} to bound the number of endpoints and recalling that, by Proposition \ref{preliminarypartitionprop}, $mesh(\eta_p(t))\leq 2/t \log \log t$, we get
\bes
%\begin{split}
Leb(\eta_{s}(t)) \geq Leb(\eta_p(t))-Leb(\Sigma_t)- \frac{2}{t \log \log t} 2 (L+1)\frac{2t}{m_f} 
%\\ &\geq  Leb(\eta_p(t)) - \delta/2 - \frac{16 L+8}{ \log \log t }\left( \frac{1}{m_f} + \frac{2}{t}\right)
%\end{split}
\ees
and, enlarging $t_1$ if necessary, both the last two terms are than $\delta/2$.

Let $t/3  \leq r \leq 2 / m_f $ and $x\in \eta_{s}(t)$. Let us show that the assumptions of Corollary \ref{BSf'cor} and Corollary \ref{growthBSf''} on the growth of $\BS{f'}{r}$ and $\BS{f''}{r}$ hold. 
By Lemma \ref{controlhbetweenrlemma}, there exists $l, l'$, with $l(t)-L \leq l \leq l(t)$ and $l'(t)-L\leq l' \leq l'(t)$ such that $h^{(n_{l})} \leq  r < h^{(n_{l + 1})}$ and $ h_\I^{(n'_{l'})} \leq  r < h_\I^{(n'_{l'+1})}$. Since by construction of $\eta_s(t)$, $x\notin \Sigma_t$, in particular, if $l<  l(t) $ and $l'<  l'(t)$, $x\notin \Sigma_l^+$, $x\notin \Sigma^-_{l'}$.
Hence in this case the assumptions (\ref{assumptionsBSf'}) of Corollaries \ref{BSf'cor} and \ref{growthBSf''} hold. In the case where $l= l(t) $ or $l'=l'(t)$, we only have $x\notin \overline{\Sigma}^+_l$ or $x\notin \overline{\Sigma}^-_{l'}$, but %one can check that 
also in this case the Corollaries hold since the only property needed in their proof is that $T^i x \notin [0,\sigma_{l(t)} \lambda^{(n_{l(t)})}]$ or $[1-\sigma_{l'(t)} \lambda_\I^{(n'_{l'(t)})}]$ for $0\leq i\leq r$ and $r \leq R_M(t)$. 

Since $r\geq t/3 \geq t_1/3$, enlarging $t_1$, one can assure that $r\geq r_o$, for the $r_o$ given by the Corollaries \ref{BSf'cor} and \ref{growthBSf''} and since $x\in I \in \eta_p(t)$, $x$ is not a singularity of $\BS{f}{r}$ by Property $1$ and $3$ of Proposition \ref{preliminarypartitionprop}.  Hence one can apply the Corollaries \ref{BSf'cor} and \ref{growthBSf''}. Moreover, by Property $3$, 
\bes
x_m \doteqdot \min_{0\leq i<r} T^i x \geq \frac{M}{t\log \log t}  \quad \mathrm{and} \quad 1-x_M \doteqdot \min_{0\leq i<r}(1- T^i x)  \geq \frac{M}{t\log \log t}. 
\ees
Thus, we get respectively
\begin{eqnarray}
 \BS{f'}{r}(x)  &\leq &
% -C_1 r\log r + \frac{C^+ (\kappa'+1)}{1-x_M} \leq 
- C_1 r\log r \left( 1 - \frac{C^+ (\kappa'+1) t\log \log t}{C_1  M r \log r}\right) ;\label{BSf'forpartition}
\\ 
| \BS{f'}{r}(x) | &\leq &
 C_2 r\log r  \left( 1 + \frac{(\kappa + \kappa' +2) t\log \log t}{ M r \log r}\right) ;\label{BSf'aboveforpartition}
\\
%\begin{split}
| \BS{f''}{r}(x) | & \leq & 
%C_4  \max \left\{\frac{1}{x_m}, \frac{1}{1-x_M} \right\} \left( r\log r + \frac{\kappa+1}{x_m} + \frac{\kappa'+1}{1-x_M} \right)  \nonumber \leq \\ &\leq & 
C_4 \frac{t \log \log t}{M} r\log r \left( 1 +     \frac{(\kappa' +\kappa+2) t\log \log t}{ M r \log r} \right) \label{BSf''forpartition} .
% \frac{\kappa}{x_m} + \frac{\kappa'}{1-x_M} \right)
%\end{split}
\end{eqnarray}
 Recalling that $t\leq 3r$ and again by enlarging $t_1$ if necessary, one can assure that the last terms in (\ref{BSf'forpartition}, \ref{BSf'aboveforpartition}, \ref{BSf''forpartition}), involving $t\log \log t / r \log r$, are less than $1/2$. Hence we get respectively (\ref{f'decreases}, \ref{SBf'boundabove}, \ref{BSf''}).%, concluding the proof of the proposition.
\end{proofof}

%Let us show that Proposition \ref{growthBSf'partition} implies also that $r(\cdot,t)$ is an decreasing function. Assume $x<y$. 
%Since in particular $\BS{f'}{r(x,t)}(x)<0$, the function $\BS{f}{r(x,t)}$ is decreasing. Hence
%\be 
%\BS{f}{r(y,t)}(x) >  \BS{f}{r(y,t)}(y) > t .
%\ee
%Using again the definition of $r(\cdot,t)$, this implies $r(x,t)< r(y,t)$.
%In particular, we justified (\ref{rarbmaxmin}) in Section \ref{geometricdescriptionsec}.
%Recalling also the notation (\ref{Deltafdef},\ref{Deltafjdef}) introduced in the same Section,

\begin{cor}\label{growthBSpartitioncor}
%There exist partial partitions $\eta_5(t)\subset \eta_4(t)$ such that 
%for each $x\in I \in \eta_5(t)$, each $r(a)\leq r \leq r(b)$,
For each $I=[a,b[\in \eta_{s}(t)$, $x\in I$ and $r(a,t)\leq r \leq r(b,t)$,
\begin{eqnarray}
&& const ( t\log t )  \leq   |\BS{f'}{r}(x)|   \leq   const' ( t\log t ) ; \label{growthf'} \\
%\BS{f'}{r_m(I)}(x) \geq const ( t\log t ).
&& \Delta f (I , t)  \geq   const \left( \frac{\log t }{\log \log t} \right) \xrightarrow{t \rightarrow + \infty } + \infty. \label{growthDeltaf}
%\\ &&|\BS{f''}{r}(x)| \leq const ( t^2\log t ). \label{growthf''}
\\
&& | \Delta f (I , t)|  =   o(\log t); \qquad  | \Delta f^1 (I , t)|  =   o(\log t) .\label{deltafologt}
% const \left( \frac{\log t }{\log \log t} \right) \xrightarrow{t \rightarrow + \infty } + \infty. \label{growthDeltaf}
\end{eqnarray}
\end{cor}
\begin{proof} Equation (\ref{growthf'}) follows from (\ref{f'decreases}) and (\ref{SBf'boundabove}) of Proposition \ref{growthBSf'partition}, since $t/3 \leq r(a,t),r(b,t) \leq 2t/m_f$. % for each $[a,b[\in \eta_{su}(t) \prec \eta_{p}(t)$.
%Consider the partial partition $\eta_5(t)$ obtained from $\eta_4(t)$ in Lemma \ref{preliminary2} by throwing away intervals completely contained in $\eta_{s}(t)$. By Lemma \ref{preliminary2}, for each $x\in I \in \eta_4(t)$, $r=r(x,t)$ satisfies the bounds in Proposition \ref{growthBSf'partition}, so (\ref{f'decreases}) holds. In particular, choosing $x_r$ so that $r=r(x_r,t)$ (possible since $r(a)\leq r \leq r(b)$,),
%$r(a,t)=\min_{x\in[a,b[} r(x,t)$,
%\be \label{growthBSf'}
%\left| \BS{f'}{r}(x)\right| \geq C  r(x_r,t) \log r(x_r,t) \geq const (t \log t).
%\left| \BS{f'}{r(a,t)}(x)\right| \geq C  r(a,t) \log r(a,t) \geq const (t \log t).
%\ee
%Let us prove (\ref{growthDeltaf}). 
Since $\BS{f}{r(a,t)}$ is continuous with its derivative on $[a,b]$ (Property 1 and 3 in Proposition \ref{preliminarypartitionprop}), by mean value theorem there exists $z\in I$ such that 
% and combining (\ref{growthf'}) with the control on the interval sizes (property 1 in Proposition \ref{preliminarypartition}, one gets
\bes
 \Delta f = \BS{f}{r(a,t)}(a) - \BS{f}{r(a,t)}(b) = - \BS{f'}{r(a,t)}(z)(b-a) \geq   \frac{const ( t\log t )}{2 t \log \log t} ,
\ees 
where we applied (\ref{f'decreases}) %(\ref{growthf'})
 and the control on the interval sizes (Property 2 in Proposition \ref{preliminarypartitionprop}).
%The bound (\ref{growthf''}) follows from (\ref{BSf''}) and the trivial upper bound $r=r(x_r,t)\leq R_M(t)$.
%\be \label{growthBSf'}
%\left| \BS{f''}{r}(x)\right| \geq C  r(x_r,t)^2 \log r(x_r,t) \geq const (t^2 \log t).
%%\left| \BS{f'}{r(a,t)}(x)\right| \geq C  r(a,t) \log r(a,t) \geq const (t \log t).
%\ee 
The proof of (\ref{deltafologt}) are obtained similarly using (\ref{growthf'}).%, using this time (\ref{}).
\end{proof}

\subsection{Mixing partitions.}\label{mainpropertiespartitionsec}
Let us prove Proposition \ref{mainpropertiespartitionprop}. 
% (Proof of Proposition \ref{mainpropertiespartitionprop}).} \label{mainpropertiespartitionsec}
%\subsection{Main Properties of the mixing partitions.} \label{mainpropertiespartitionsec}
%%%In this Section we prove Proposition \ref{mainpropertiespartitionprop} by refining $\eta_{s}(t)$ to get the final mixing partitions $\eta_{m}(t)$. 
%%%%ELIMINABILE
\paragraph{Outline.}
The uniform vertical distribution (\ref{uniformstretch}) is proved in Section \ref{uniformstretchsec}.
In order to show that $J(I,t)$
%, number of curves generated from each $\varphi_t(I)$, 
is tending to infinity and moreover that it is asymptotic to $\Delta f$ (i.e. (\ref{asymptoticscurves})), we need a further refinement of the partitions that guarantees that, exploiting ergodicity, the number of fibers covered in time $\Delta f$ is asymptotic to $\Delta f$, since $\int_{I^{(0)}}f(x)\ud x =1$.

The refinement is constructed in two steps. In Section \ref{upperlowerboundsec} we prove a first rough upper bound (Corollary \ref{Jupperboundlemma}), which allows us to refine the partitions in order to have a better control of the distance from singularities for $r(a)\leq r \leq r(a)+1$, i.e. in the curves range (Lemma \ref{curvesdistsinglemma}). Using this refinement one can prove the control on the variation of the curves slopes (Section \ref{variationslopessec}).

In Section \ref{asymptoticscurvesandequidistributionsec}, the same refinement is used to give a lower bound on the number of curves and hence to construct a second refinement of the partitions on which one can get by ergodicity both equidistribution along the base (\ref{equidistributionbase}) and the curve asymptotics (\ref{asymptoticscurves}).% (Section \ref{asymptoticscurvesandequidistributionsec}).


\subsubsection{Uniform vertical distribution.}\label{uniformstretchsec}
%%%ELIMINABILE
Let us show that given $\epsilon >0$, choosing $M>M_0(\epsilon)$, each $I\in \eta_{s}(t)$ satisfies Property $1$. % (\ref{uniformstretch}), i.e. 
%\bes
%\left|\frac{ h(R)(b-a) }{\Delta f^j} - Leb(I^{h(R)}_j) \right| \leq \frac{ h(R)(b-a) }{\Delta f^j} \epsilon  
%\ees
%
Let us recall the following definition used in \cite{Fa:ana} (see also \cite{Ko:mix,Ko:nonI,Ko:nonII}).
\begin{defn}
Given $\epsilon>0$ %and $K>0$
, the function $g$ on the interval $[a,b]$ is \emph{
%$(\epsilon,K)-$
$\epsilon$-uniformly distributed} if 
%\bes
%\sup_{[a,b[}g - \inf_{[a,b[}g \geq K,
%\ees and 
for any $c,d$ such that $\inf_{[a,b[}g \leq c \leq d \leq \sup_{[a,b[}g$,
%\be \inf_{[a,b[}g \leq c \leq d \leq \sup_{[a,b[}g,
%\label{cdbounds}
%\ee
%$\inf_{[a,b[}g \leq c \leq d \leq \sup_{[a,b[}g$,
the measure of the set 
%%%%\bes
$I_{c,d} = \{x \in [a,b[ \, | \quad c\leq g(x) \leq d \}$ 
%\ees
%has measure
satisfies
\be \label{measIcd}
(1-\epsilon) \frac{ d-c  }{\sup_{[a,b[}g - \inf_{[a,b[}g}  \leq \frac{Leb(I_{c,d})}{b-a} \leq (1+\epsilon) \frac{ d-c  }{\sup_{[a,b[}g - \inf_{[a,b[}g} .
\ee
\end{defn}

%%%%%%%%%%%%%Puoi probabilmente eliminare il K, basta la seconda condizione.
In \cite{Fa:ana} Fayad proves the following criterium for uniform distribution.
\begin{lemma}[Fayad]\label{Fayadunifstretchlemma}
If $g$ is monotonic and 
%\be
%\frac{\sup_{[a,b[}|g''(x)||b-a|}{\inf_{[a,b[}|g'(x)| } \leq \epsilon,
%\ee
\be
\sup_{[a,b[}|g''(x)||b-a| \leq \epsilon \inf_{[a,b[}|g'(x)| ,
\ee
then $g$ is $\epsilon$-uniformly distributed on $[a,b]$.
\end{lemma}

For each $[a,b[ \, \in \eta_{s}(t)$, consider $\BS{f}{r(a)+j}$, for $j=1,\dots, J([a,b[,t)-2$. % extended by continuity to $[a,b]$.
%By (\ref{growthf'}) and the control on the interval sizes (Property 2, Lemma \ref{preliminary1}),
%\bes
%\inf_{[a,b[}|\BS{f'}{r(a)+j-1}(x)| |b-a|\geq \frac{ const (t \log t)}{2t \log \log t} \doteqdot K(t) \xrightarrow{t\rightarrow + \infty} + \infty.
%\ees
%By the control of the interval sizes (
From Property 2 of Proposition \ref{preliminarypartitionprop}, (\ref{f'decreases}, \ref{BSf''}) of Proposition \ref{growthBSf'partition} and $r\leq 2t/m_f$,
%Using also (\ref{growthf''}),
\bes \begin{split}
\frac{\sup_{[a,b[}|\BS{f''}{r(a)+j}(x)||b-a|}{\inf_{[a,b[}|\BS{f'}{r(a)+j}(x)| } 
%&
\leq \frac{  C'' r^2 (\log r) ( \log \log r )(b-a)  }{M  C_1' r \log r }  \leq \\
%&
\leq \frac{C'' (\frac{2t}{m_f} \log \log \frac{2t}{m_f})  \frac{2}{  t \log \log t}}{M C_1'}   \xrightarrow{t\rightarrow + \infty}  \frac{4 C'' }{ M C_1' m_f} .
     \end{split}
\ees
Choosing  $M > M_0(\epsilon ) \doteqdot 8 C''/ C_1' m_f \epsilon$ and then $t_3\geq t_2$ large enough, the last expression is less than $\epsilon$ for $t\geq t_3$.
Hence, each $\BS{f}{r(a)+j}$, for $j=1,\dots, J\!-\! 2$, being also decreasing, is $\epsilon$-uniformly distributed on $[a,b[\, \in \eta_{s}(t)$ by Lemma \ref{Fayadunifstretchlemma}.
The set $I_{j}^{h(R)}$ defined in (\ref{Ijhdef}) is of the form $I_{c,d}$ for $g=\BS{f}{r(a)+j}$, $c= t -h(R)$ and $d=t$; by the definition (\ref{splittingpointsdef}) of splitting points, one can check that $d=g(y_{j})\leq \sup_{[a,b[} g $ and $c \geq g(y_{j+1})\geq \inf_{[a,b[} g$. 
%, using the definition of splitting points (\ref{splittingpointsdef}), can be written as 
%\bes
%\begin{split}
%I_{j}^{h(R)} & \doteqdot \{ x | \quad t - h(R)  \leq \BS{f}{r(a)+j-1}(x) \leq t  \} = \\ 
% &=  \{ x | \quad \BS{f}{r(a)+j-1}(y_j) - h(R)  \leq \BS{f}{r(a)+j-1}(x) \leq \BS{f}{r(a)+j-1} (y_j) \},
%\end{split}
%\ees
%hence it is of the form $I_{c,d}$ for $g=\BS{f}{r(a)+j-1}$, $c= \BS{f}{r(a)+j-1}(y_j) -h(R)$ and $d=\BS{f}{r(a)+j-1} (y_j)$. 
Hence, by (\ref{measIcd}), its measure is bounded by
\be
(1-\epsilon ) \frac{h(R)}{\Delta f^j ([a,b[,t) } (b-a)\leq  Leb(I_{j}^{h(R)} ) \leq (1+\epsilon ) \frac{h(R)}{\Delta f^j ([a,b[,t) } (b-a) .\ee
%Choosing $t\gg 1$, so that $\epsilon(t)< \epsilon $, 
This proves the uniform vertical distribution Property (\ref{uniformstretch}).

%\subsubsection{Rough upper and lower bound on the number of curves.}\label{upperlowerboundsec}
%Recall that the number of curves generated from each $\varphi_t(I)$  is given by $J(I,t)=r(b)-r(a)+1$. An equivalent characterization, as proven in the next paragraph, is the following.
%\begin{lemma}\label{equivalentJlemma}
%\be\label{equivalentJ}
%J(I,t)-1= r(b)-r(a) =r\left( T^{r(a)} b  , \Delta f(I,t) \right). 
%\ee
%\end{lemma}
%Lemma \ref{equivalentJlemma} shows that the number of strips is determined by the number of fibers that the point $T^{r(a)}b$ still has to cover when $a$ stops, for the delayed time $\Delta f$ accumulated through the stretching of $\BS{r}{f}$.

%\paragraph{An equivalent definition of $J(I,t)$.}
%\begin{proofof}{Lemma}{equivalentJlemma}
%Let us remark the following relation between Birkhoff sums. For each $s\in \mathbb{N}$,
%\be\label{BSrel}
%\BS{f}{s+r(a)} (b)= \BS{f}{r(a)} (b) + \BS{f}{s} (T^{r(a)}b).
%\ee
%Applying it to $s = r(b)-r(a)$ and using the definition of $r(a)$ and $r(b)$,
%\bes
%\BS{f}{r(b)-r(a)} (T^{r(a)}b) > t -  \BS{f}{r(a)} (b) >  \BS{f}{r(a)} (a)  -  \BS{f}{r(a)} (b) = \Delta f.
%\ees
%Hence, again by definition of $r(\cdot,t)$, we get $r\left( T^{r(a)b } , \Delta f \right) \leq r(b)-r(a)$.

%Conversely, apply (\ref{BSrel}) to $s=r( T^{r(a)b } , \Delta f(I,t)) + r(a)$.
%\bes
%\begin{split}
% \BS{f}{ r ( T^{r(a)} b  , \Delta f)  + r(a) )}(b) = & \BS{f}{r(a)} (b) + \BS{f}{r( T^{r(a)} b ) , \Delta f)} (T^{r(a)}b) >\\ & \BS{f}{r(a)} (b) + \Delta f = \BS{f}{r(a)} (a) > t.  
%\end{split}
%\ees
%Hence, by definition of $r(\cdot,t)$, we get $r(b)<r ( T^{r(a)} b  , \Delta f  ) + r(a) $, which is the opposite inequality.
%\end{proofof}

 %\subsubsection{A Denjoy-Kosma type of argument.}
 %Remark that the classical Denjoy-Kosma argument for the rotation works in the following setting. Let us denote by $\Var{f}{I}$ the variation of the function $f$ on $I$.

% \begin{lemma}\label{classDKlemma}
% Let $n$ be a Zorich step and let $r< h^{(n)}_j$. Assume that $x,y \in T^{k}I^{(n)}_{j}$ belong to the same floor of $Z^{(n)}_j$ for some $0 \leq k < h^{(n)}_{j}-r$. Assume moreover that $T^i[x,y] \cap I' = \emptyset$ for some interval $I'$ and each $0\leq i\leq r$. Then 
% \be \label{classDK}
% \left | \BS{r}{f}(x) - \BS{r}{f}(y) \right| \leq\Var{f}{I^{(0)}\backslash I'}.
% \ee 
% \end{lemma}
% \begin{proof}
% Remark that in particular $\Orb{r}{x} \prec Z^{(n)}_{j}$ and $\Orb{r}{y} \prec Z^{(n)}_{j}$ and hence $T^{i}x$ and $T^i y$ belong to the same floors of $Z^{(n)}_j$ for each $0\leq i < r$. Since floors are disjoint, $T^i [x,y]$ do not overlap for each $0\leq i < r$. 
% Moreover, by assumption $\cup_{i=0}^r T^i [x,y] \subset I^{(0)}\backslash I' $.
% Hence (\ref{classDK}) follows from the usual Denjoy-Kosma argument. 
% \end{proof}

% Remark that it is essential than $x,y$ belong to the same floor. Also, if $\Orb{r}{x}$ (and automatically $\Orb{r}{y}$) is not completely contained in the same tower, $T^i [x,y]$ are not necessarily disjoint and hence one cannot apply this type of argument. %

% In order to estimate the variation of Birkhoff sums on each partition interval $I \in \eta_{s}(t)$, the idea is to decompose each $I$ into  
% floors of some Zorich step on which one can apply Lemma \ref{classDK} and to relate Birkhoff sums on each of them by continuity. The result is given in Lemma \ref{varBSbyDKlemma} below.
% In order to assure an upper bound on the number of floors in which $I$ is decomposed, one needs balance and a diophantine condition to estimate the gap between balanced steps. 

% We will use the following diophantine condition, proved by Khanin and Nogueira using a result by Kerhoff.
% \begin{prop}\label{Khanin}
% For $\mu_{\Z}$-a.e. IET $T$ there exists $\overline{\nu}$, $\overline{\kappa}$ and $\gamma>0$ such that one can find an infinite subsequence $\{m_l \}_{l\in \mathbb{N}}$ of Zorich times such that lenghts and heights are balanced, i.e. 
% \bes
% \frac{1}{\overline{\nu}} \leq \frac{\lambda^{(m_l)}_i}{\lambda^{(m_l)}_j} \leq \overline{\nu} ; \qquad \frac{1}{\overline{\kappa}} \leq \frac{h^{(m_l)}_i}{h^{(m_l)}_j} \leq \overline{\kappa}; \qquad \forall 1\leq i,j \leq d ,
% \ees
% and $\ZLp{m_l,m_{l+1}}>0.$
% Furthermore, recalling that $h^{(m_l)} \doteqdot \max_{i=1,\dots, d} h^{(m_l)}_i$,
% \be \label{Khanindistance}
% \frac{h^{(m_{l+1})}}{h^{(m_{l})}} \leq C l^{\gamma}.
% \ee
 %\end{prop}

% Lemma \ref{classDK} and Proposition \ref{Khanin} are used to prove the following.
% \begin{lemma}\label{varBSbyDKlemma}
% Assume that $T$ satisfies Proposition \ref{Khanin}. Let $I\in \eta_{s}(t)$. For each $x,y\in I$ and $0\leq r \leq r_M(t)$, 
% \be \label{varBSbyDK}
% \left| \BS{r}{f}(a) - \BS{r}{f}(b) \right| = o\left( (\log t) ^{\gamma + 3} \right).
% \ee
% \end{lemma}
% \begin{proof}
% Let $\{m_l\}_{l\in \mathbb{N}}$ the sequence of times in Proposition \ref{Khanin}.
% Choose $\overline{l} = \overline{l} (t)$ such that 
% \bes 
% m_{\overline{l}} \doteqdot \min \{ m \in \mathbb{N} | \, \lambda^{(m)} \leq \frac{1}{t \log t} \}.
% \ees
% Recall that by Property 3 in Proposition \ref{preliminarypartitionprop}, for each $I\in  \eta_{s}(t)$,
% \bes
% T^i I \cap \left[0, \frac{1}{t \log t} \right] \cap \left[ 1-\frac{1}{t \log t} ,1 \right] = \emptyset , \quad 0\leq i \leq R_M .
% \ees
% In particular, since by our choice of $\overline{l}$, $I^{(m_{\overline{l}})} \subset [0,1/(t \log t)]$, for each $x\in I\in \eta_p(t)$, $T^i x \notin I^{(m_{\overline{l}})}$ for each $0\leq i \leq R_M(t)$, i.e. $x$ does not belong to the top $R_M$ floors of the towers $Z^{(m_{\overline{l}})}_j$. So we also get $\Orb{x}{r(x,t)} \prec Z^{(m_{\overline{l}})}_{j_0}$ for some ${j_0}$.

% Since the lenght of $I=[a,b[$ is bigger than $\lambda^{(m_{\overline{l}})}$, two points $x,y \in I$ do not belong to the same floor, hence we cannot apply Lemma \ref{classDKlemma} directly. Consider $\xi_{m_{\overline{l}}} (I)$, partition of $I$ into floors of step $m_{\overline{l}}$ and denote the endpoints of such a partition by
% \bes
% x_0 \doteqdot a < x_1 < \dots < x_i < \dots < x_{F-1} < x_F \doteqdot b, \qquad [x_i,x_{i+1}[ \in \xi_{m_{\overline{l}}} (I). 
% \ees
% We just proved that for each $x,y \in [x_i,x_{i+1}[$ all the assumptions of Lemma \ref{classDK} hold when $n=m_{\overline{l}}$ for $I\backslash I' = \left] \frac{1}{t \log t} , 1- \frac{1}{t \log t} \right[ $ and $r\leq R_M$, hence
% \be\label{DKcontrolxyinF}
% \left | \BS{r}{f}(x) - \BS{r}{f}(y) \right| \leq\Var{f}{ \left] \frac{1}{t \log t} , 1- \frac{1}{t \log t} \right[} = O (\log (t \log t)).   \ee 

% In order to estimate the number $F$ of floors, let us find a lower bound for $\min_j \lambda^{(m_{\overline{l}})}_j$. By definition of $m_{\overline{l}}$, $\lambda^{(m_{\overline{l}-1})} > 1/(t\log t)$. Exploiting the balance and (\ref{Khanindistance}) properties in Proposition \ref{Khanin} and the trivial fact that $\lambda^{(m_{l})} \min_{j}h^{(m_{l})}_j \leq 1 \leq \lambda^{(m_{l})}\max_{j}h^{(m_{l})}_j $,
% \be \label{ratiolambdas}
% \frac{\lambda^{(m_{\overline{l}})}}{\lambda^{(m_{\overline{l}-1})}} \geq  \frac{1/\max_{j}h^{(m_{\overline{l}})}_j }{ 1/\min_{j}h^{(m_{\overline{l}-1})}_j }\geq  \frac{h^{(m_{\overline{l}-1})}}{\overline{\kappa} h^{(m_{\overline{l}})}} \geq \frac{1}{\overline{\kappa} C (\overline{l}-1)^{\gamma}}.
% \ee 
% By positivity of the matrices $\ZLp{m_l,m_{l+1}}$, $ h^{(m_{\overline{l}-1})} \geq d^{\overline{l}-1}$, which by balance translates into $\overline{\kappa}/ d^{\overline{l}-1} \geq \lambda^{(m_{\overline{l}-1})} \geq  1/(t\log t)$ and hence gives $\overline{l}-1 \leq const \log (t\log t) $.
% Together with lenghts balance and (\ref{ratiolambdas}) we get
% \bes
% \min_j \lambda^{(m_{\overline{l}})}_j \geq \frac{1}{\overline{\nu}}\lambda^{(m_{\overline{l}})} \geq  \frac{\lambda^{(m_{\overline{l}-1})}}{\overline{\nu} \overline{\kappa} C (\overline{l}-1)^{\gamma}} \geq  O \left(  \frac{1}{ t \log t (\log (t\log t) )^{\gamma}}\right) .
% \ees
% Going back to the number of floors, using the lower bound on the size of $I$ in Property 2 of Proposition \ref{preliminarypartitionprop},
% \be \label{boundF}
% F \leq \frac{ Leb{(I)}}{\min_j \lambda^{(m_{\overline{l}})}_j } + 1 \leq O \left( \frac{ t \log t (\log (t\log t))^{\gamma} }{ t \log \log t} \right) = o \left( (\log t )^{\gamma + 2}  \right) .
% \ee

% For each fixed $t>0$, remark that $\BS{r}{f}$ is continuous on $I\in \eta_{s}(t)$ for $r\leq R_M(t)$, since both $T^j$, $j\leq  R_M(t)$ and $f$ are continous on $I$ by Properties 1 and 3 in Proposition \ref{preliminarypartitionprop}. Hence one can find points $x_{i}^t$ sufficiently close to each $x_{i}$ for $i=0,\dots, F$, with $x_{i-1} \leq  x_i^t \leq x_i$, such that
% \be\label{controlnearendpoints}
% \left | \BS{r}{f}(x_i^t) - \BS{r}{f}(x_i) \right| \leq \log t .
% \ee
% Let us decompose in a telescopic sum
% \be\label{telescopicdecomp}
% \begin{split}
% \left| \BS{r}{f}(a) - \BS{r}{f}(b) \right|  = \left| \sum_{i=0}^{F-1} \BS{r}{f}(x_{i+1}) -  \BS{r}{f}(x_{i}) \right| \leq \\  \leq 
 %  \sum_{i=0}^{F-1}  \left| \BS{r}{f}(x_{i}) -  \BS{r}{f}(x_{i+1}^t) \right| +  \sum_{i=0}^{F-1}  \left| \BS{r}{f}(x_{i+1}^t) -  \BS{r}{f}(x_{i+1}) \right| .
 %\end{split}
 %\ee
 %Since $x_i,x_{i+1}^t$ both belong to the same floor $[x_i, x_{i+1}[ \in \xi_{m_{\overline{l}}} (I)$, we can apply (\ref{DKcontrolxyinF}) to control each of the terms in the first sum in the RHS of (\ref{telescopicdecomp}), while (\ref{controlnearendpoints}) controls the terms in the second sum. Hence, 
% \bes
% \left| \BS{r}{f}(a) - \BS{r}{f}(b) \right| \leq F \, O(\log t) =
% o(\log t ^{ \gamma +3} ),
% \ees
% by using (\ref{boundF}) to control $F$.
% \end{proof}

\subsubsection{Rough upper and lower bound on the number of curves.}\label{upperlowerboundsec}
%%%ELIMINABILE Anche una riga nell'enunciato del lemma cambiando referenze 
%MEttere invece: The number of curves generated from each $\varphi_t(I)$ can be estimated as follows. 
Recall that the number of curves generated from each $\varphi_t(I)$  is given by $J(I,t)=r(b,t)-r(a,t)+1$. 
\begin{lemma}\label{equivalentJlemma} For each $I=[a,b[$,
\be\label{equivalentJ}
r\left( T^{r(a)}b,\Delta f\right) + 1 \leq J(I,t)\leq  r\left( T^{r(a)+1}b,\Delta f^1\right) + 3.
%J(I,t)-1= r(b)-r(a) =r\left( T^{r(a)} b  , \Delta f(I,t) \right). 
\ee
\end{lemma}
%%%%ELIMINABILE
Lemma \ref{equivalentJlemma} shows that the number of strips is related to the number of fibers that the point $T^{r(a)}b$ still has to cover in time $\Delta f$  when $a$ stops, because of the delay accumulated through the stretching of $\BS{f}{r(a)}$.

%In order to show that $J(I,t)$
%, number of curves generated from each $\varphi_t(I)$, 
%is tending to infinity and moreover that it is asymptotic to $\Delta f$ (see Asmptotics of curves (\ref{asymptoticscurves})), we need a further refinement of the partition that guarantees that, exploting ergodicity, the number of fibers covered in time $\Delta f$ is proportional to $\Delta f$, since $\int_{I^{(0)}}f=1$.

%The refinement will be constructed in two steps. In this section we prove a first rough upper bound (Lemma \ref{Jupperbound}), based on a Denjoy-Kosma type of argument (Lemma \ref{varBSbyDKlemma}). The upper bound allows us to refine the partition to have a better control of the distance from singularities in the curves range, i.e. $r(a)\leq r \leq r(a)+1$ (Lemma \ref{curvesdistsinglemma}). This control allows to give a lower bound on the number of curves and hence construct the partition on which one can get the precise asymptotics by ergodicity (Section \ref{asymptoticscurvesandequidistributionsec}).

%\paragraph{An equivalent definition of $J(I,t)$.}
\begin{proof}%of}{Lemma}{equivalentJlemma}
Applying the relation $\BS{f}{r_1+r_2}(x)=\BS{f}{r_1}(x)+\BS{f}{r_2}(T^{r_1}x)$ and the definition (\ref{defr}) of $r(\cdot,\cdot)$,
%By a trivial relation between Birkhoff sums and using the definition of $r(\cdot , \cdot)$,
%Let us remark the following relation between Birkhoff sums. For each $s\in \mathbb{N}$,
%\be\label{BSrel}
%\BS{f}{s+r(a)} (b)= \BS{f}{r(a)} (b) + \BS{f}{s} (T^{r(a)}b).
%\ee
%Applying it to  $s=r( T^{r(a)b } , \Delta f(I,t))$ and using the definition of $r(\cdot , \cdot)$,
\bes
\begin{split}
 \BS{f}{ r ( T^{r(a)} b  , \Delta f)  + r(a) }(b)& =  \BS{f}{r(a)} (b) + \BS{f}{r( T^{r(a)} b ) , \Delta f)} (T^{r(a)}b) \leq \\ & \leq \BS{f}{r(a)} (b) + \Delta f = \BS{f}{r(a)} (a) \leq t.  \end{split}
\ees
Hence, $r(b,t) \geq r ( T^{r(a)} b  , \Delta f  ) + r(a,t) $, which is the first inequality in (\ref{equivalentJ}).

%%%%ELIMINABILE The second is analogous.
For the second inequality, 
\bes
\begin{split}
 \BS{f}{ r ( T^{r(a)+1 } b  , \Delta f^1)  + r(a) +2 }(b)& =  \BS{f}{r(a)+1} (b) + \BS{f}{r( T^{r(a)+1} b  , \Delta f^1 )+1 } (T^{r(a)+1}b)  \\ & > \BS{f}{r(a)+1} (b) + \Delta f^1 = \BS{f}{r(a)+1} (a) > t, 
\end{split}
\ees
which implies that $r(b)<  r ( T^{r(a)+1 } b  , \Delta f^1)  + r(a) +2 $.
\end{proof}

%%%%ELIMINABILE Titoletti
\paragraph{Rough upper bound on the number of curves.}
\begin{cor} %[Rough upper bound on $J(t)$]
\label{Jupperboundlemma}
Let $J(t)\doteqdot \sup_{I\in \eta_{s}(t)} J(I,t)$. Then
%There exists $\Gamma>0$ such that
%%%%ELIMINABILE una riga cambiando ref
\be\label{Jupperbound}
J(t) =  o(\log t).
\ee
\end{cor}
\begin{proof}
By Lemma \ref{equivalentJlemma},
%%%%\bes
$J(I,t) \leq   r ( T^{r(a)+1 } b  , \Delta f^1) +3\leq \frac{\Delta f^1(I,t)}{m_f} +3$. 
%\ees
Recalling that $\Delta f^1(I,t) = o\left( \log t \right)$  by (\ref{deltafologt}) of Corollary \ref{growthBSpartitioncor} we get the bound.
\end{proof}

\paragraph{Pull-back using discrete times.}
Consider the map $R_t: I^{(0)}\rightarrow  I^{(0)}$ given by $R_t(x)= T^{r(x,t)}x $, which is the projection of $ \varphi_t(x,0)\in X_f$ to the base $I^{(0)}$.
%$=P_{ I^{(0)}} \cdot \varphi_t(x,0)$, where $P_{ I^{(0)}}$ is the projection of $X_f$ on $I^{(0)}$. In particular,
%\bes
%\mathrm{if} \, \varphi_t(x,0)= (T^{r(x,t)}x,t-\BS{f}{r-1}(x)), \qquad  R_t(x) = T^{r(x,t)}x .  
%\ees
Remark that $R_t$ in general is not one to one.
The following lemma is used by Kochergin in \cite{Ko:mix} (Lemma 1.3).
\begin{lemma}[Kochergin]
For any measurable set $S \subset I^{(0)}$,
\be \label{Kochlemma}
Leb(R_t^{-1} S) \leq \int_S \left( \frac{f(x)}{m_f} +1 \right) \ud x.
\ee
\end{lemma}
Since $f\in L^1$, by absolute continuity of the integral, for any $\delta>0$ it is possible to choose $\delta_1$ such that the RHS of (\ref{Kochlemma}) is bounded by $\delta$ as long as $Leb(S)< \delta_1$. Hence we get the following corollary.
\begin{cor}\label{Kochlemmacor}
For each $\delta >0$, there exists $\delta_1>0$ such that for any measurable $S \subset I^{(0)}$ , if $Leb(S)< \delta_1$, then
$Leb(R_t^{-1} S) < \delta $.
\end{cor}

\paragraph{Refinement of the partitions.}
%%%%%ELIMINABILE Let us refine the partitions to have also a lower bound.
We want to refine the partitions in order to throw away points $x$ such that when considering $r$ such that $r(a)\leq r \leq r(a)+ J$, i.e. in the range of the curves, the distance of $T^{r}x$ from singularities is bounded from below by $1/\log t$. %^\Gamma$ for some $\Gamma>0$.
\begin{lemma}\label{curvesdistsinglemma}
There exist partitions $\eta_{5}(t)\subset \eta_{s}(t)$ and $t_3\geq t_2$ such that, for $t\geq t_3 $, $Leb(\eta_5(t))\geq Leb(\eta_{s}(t))-\delta$ and for each $x\in I \in \eta_5(t)$, 
\be \label{curvesdistsing}
|T^r x| \geq \frac{1}{ (\log t)^2 };\quad |1- T^r x| \geq \frac{1}{(\log t)^2 },
%|T^r x| \geq \frac{1}{(\log t)^\Gamma };\quad |1- T^r x| \geq \frac{1}{(\log t)^\Gamma },
\ee
for each $r(a,t)\leq r \leq r(a,t)+J(I,t)$.
\end{lemma}
\begin{proof}
Define 
%Let $\Gamma$ be the same than in Lemma \ref{Jupperboundlemma}.
\be
U_3(t) \doteqdot \bigcup_{i=-[J(t)]}^{[J(t)]} T^i \left( \left[ 0, \frac{1}{(\log t)^2} \right] \right)  \cup  \bigcup_{i=-[J(t)]}^{[J(t)]} T^i \left( \left[ 1- \frac{1}{(\log t)^2} ,1 \right] \right).
\ee
%\bes
%Leb(U_3(t))\leq 4J(t) \left( \frac{1}{(\log t)^{\Gamma}} \right) \xrightarrow{t \rightarrow + \infty } 0.
%\ees
Since the continuity intervals for $T^{[J(t)]}$ and  $T^{-[J(t)]}$ are at most $d([J(t)]+1)$ (see Section \ref{continuitysec}),  
the set $U_3(t)$ consists of at most $O(J(t)^2)$ disjoint intervals. 
Consider the $2/t\log\log t$-neighbourhood of $U_3(t)$, i.e. let $U_4(t)\doteqdot \{ \, x \in I^{(0)}|\,\, d(x, U_3(t))\leq 2/t\log\log t \, \}$. 
%Denote them by $U_3(t) = [s^{1}_a, s^{1}_b] \cup \dots \cup [s^{K}_a, s^{K}_b] $, with $K\leq O(J(t)^2)$. Let us add to each of them \emph{pillow intervals} of length $2/(t \log \log t)$:
%\be
%U_4(t) \doteqdot \bigcup_{i=1}^{K} \left[ s^{i}_a - \frac{2}{t \log \log t}, s^{i}_b  +  \frac{2}{t \log \log t}\right].
%\ee
Hence, using Corollary \ref{Jupperboundlemma}, 
\bes
Leb(U_4(t))\leq \frac{4J(t)+4}{(\log t)^2} + \frac{O(J(t)^2)}{t \log \log t}
%O(J(t)^2) \left( \frac{1}{(\log t)^2} +  \frac{4}{t \log \log t}  \right) 
\xrightarrow{t \rightarrow + \infty } 0.
\ees
Choosing $t_3>t_2$ so that for $t\geq t_3$,  $Leb(U_4) < \delta_1$ where $\delta_1$ is given by Corollary \ref{Kochlemmacor}, we get $Leb(R_t^{-1}(U_4) )< \delta$. Define a refined partition $\eta_5(t)\subset \eta_s(t)$ by
\bes
 \eta_5(t) \doteqdot  \eta_{s}(t) \, \backslash \, \{ I  \in \eta_{s}(t) | \quad I \subset  R_t^{-1} U_4(t) \}.
\ees
Clearly $Leb(\eta_5(t))\geq Leb(\eta_{s}(t)) - \delta$.
%%%%
Let us show that for each $I\in  \eta_5(t)$, we get (\ref{curvesdistsing}). 
By construction there exists $x\in I$ such that $R^t (x) = T^{r(x,t) }x \notin  U_4(t)$. Hence, by Proposition \ref{preliminarypartitionprop}, $T^{r(x,t) }y \notin  U_3(t)$ for each $y\in I$.  
%Thanks to the pillow intervals,  $T^{r(x,t) }y \notin  U_3(t)$ for each $y\in I$. Indeed, consider the case $x<y$ and assume by contradiction that $T^{r(x,t) } y \in [s^{i}_a,s^{i}_b] \subset U_3(t)$. Hence, since $T^{r(x,t)}$ is an orientation preserving isometry on $I$, being continuous by Property 1 of Proposition \ref{preliminarypartitionprop}, %in Lemma \ref{preliminary1}, 
%\bes
%T^{r(x,t) }x =  T^{r(x,t)} y - ( T^{r(x,t)} y -T^{r(x,t) }x ) =  T^{r(x,t)} y - (y-x) > s^{i}_a - \frac{2}{t \log \log t} ,
%\ees
%hence, since also $T^{r(x,t)}x<T^{r(x,t) }y<s^{i}_b$, we proved that $x\in U_4(t)$, which is the contradiction.
%%%
For each $r=r(a),\dots,r(a)+ J$, the point $T^r y$ satisfies the inequalities (\ref{curvesdistsing})
by definition of $U_3(t)$, because $T^{r(x,t)}y \notin U_3(t)$, as shown above, and  $|r(x,t)-r|\leq J$.
% for each $j=1,\dots, J(t)$, $T^r(y,t)$ satisfies the inequalities (\ref{curvesdistsing})
\end{proof}

%\paragraph{Rough lower bound on the number of curves.}
%Lemma \ref{curvesdistsinglemma} allows to give also a rough lower bound on $J(t)$.
\begin{lemma}[Rough lower bound on $J(t)$]\label{roughlowerboundJlemma} 
Let $J(t)\doteqdot \sup_{I\in \eta_{5}(t)} J(I,t)$.
%For each $I\in \eta_5(t)$
\be\label{roughlowerboundJ} 
J(t)\geq O\left( \frac{\log t}{ (\log \log t)^2}\right) \xrightarrow{t\rightarrow + \infty} + \infty.
\ee
\end{lemma}
\begin{proof}
%By Lemma \ref{equivalentJlemma}, $J(I,t)\geq r ( T^{r(a)} b  , \Delta f )+1$.
For each $I=[a,b[\, \in \eta_5(t)$, by Lemma \ref{curvesdistsinglemma}, $f(T^{r}b) \leq O(  \log (\log t))$ for each $r(a,t) \leq r \leq r(a,t) + J(I,t)$. Hence, since by Lemma \ref{equivalentJlemma}, $J(I,t)\geq r ( T^{r(a)} b  , \Delta f )+1$, 
\bes 
J(I,t)  \geq  \frac{\BS{f}{ r ( T^{r(a)}b  , \Delta f)+1 }(T^{r(a)}b )}{\max_{0\leq i <  r ( T^{r(a)}b  , \Delta f)+1 }  f(T^{r(a)+i}b)} \geq \frac{\Delta f}{O( \log (\log t))},
%{\max_{r(a) \leq r \leq r(a)+ J(I,t)}  f(T^r b)} \geq  \frac{ O\left( \frac{\log t }{\log \log t} \right) }{O( \log (\log t)) },
\ees 
%\bes \begin{split}
%J(I,t)& \geq  r ( T^{r(a)}b  , \Delta f)+1  \geq  \frac{\BS{f}{ r ( T^{r(a)}b  , \Delta f)+1 }(T^{r(a)}b )}{\max_{0\leq i <  r ( T^{r(a)}b  , \Delta f)+1 }  f(T^{r(a)+i}b)} \geq
%\\&\geq  \frac{\Delta f}{\max_{r(a) \leq r \leq r(a)+ J(I,t)}  f(T^r b)} \geq  \frac{ O\left( \frac{\log t }{\log \log t} \right) }{O( \log (\log t)) },
%     \end{split}
%\ees 
which gives (\ref{roughlowerboundJ}) by using the bound (\ref{growthDeltaf}) on $\Delta f$. 
\end{proof}

\subsubsection{Variation of slopes.}\label{variationslopessec}
%Let us prove that for each
Given $I=[a,b[ \in \eta_5(t)$ the variation of the average slope of the curves $(t- \BS{f}{r(a)+j})|_{I}$, for $0\leq j\leq J(t)$ can be written as
%, is controlled by (\ref{variationslopes}), i.e.
%\bes
%\left| \frac{\Delta f}{\Delta f^j} -1 \right|    \leq  \epsilon, \quad j=1,\dots, J.
%\ees
%Let us rewrite the numerator as
\bes
\begin{split}
\left| \Delta f^j - \Delta f \right| & =
%}{\Delta f} = \frac{ 
\left| \BS{f}{r(a)+j }(a) -  \BS{f}{r(a)+j }(b) -  \BS{f}{r(a)}(a) + \BS{f}{r(a)}(b) \right| \\
%}{ \BS{f}{r(a) }(b) -  \BS{f}{r(a) }(a) } 
&= \left| \sum_{i=0}^{j-1} f(T^{r(a)+i} a)  -  \sum_{i=0}^{j-1} f(T^{r(a)+i} b ) \right| \leq   \sum_{i=0}^{j-1} \left| f'(T^{r(a)+i}c)\right|  (b-a) ,
%}{ \BS{f}{r(a) }(b) -  \BS{f}{r(a) }(a) }
\end{split}
\ees
where in the last estimate $a\leq c \leq b$ by mean value theorem. % as in the proof of Corollary \ref{growthBSpartitioncor}.% we used again the mean value theorem and $ a \leq c\leq T^{r(a)} b$.
%
Using Lemma \ref{curvesdistsinglemma}, $|f'(T^{r(a)+i}c)| \leq O \left( (\log t)^2 \right) $ for each $0\leq i<j \leq  J(t)$. 
Hence, applying also the bound on $J(t)$ given by Corollary \ref{Jupperboundlemma}, the growth estimate (\ref{growthDeltaf}) for $ \Delta f$ and the size control of $(b-a)$ (Property 2 of Proposition \ref{preliminarypartitionprop}),
%%%ELIMINARE sostituendo con quello sotto
%Hence, by Corollary \ref{Jupperboundlemma}, (\ref{growthDeltaf}) and Property 2 of Proposition \ref{preliminarypartitionprop}),
\bes
\left| \frac{\Delta f^j - \Delta f}{\Delta f} \right| \leq \frac{ J(t) \sup_{i=0}^{J-1} |f'(T^{r(a)+i}c)| (b-a)}{ \Delta f  } \leq  \frac{ o(\log t ) O \left( (\log t)^2 \right) }{ O \left( \frac{\log t }{\log \log t} \right) t \log \log t  } ,
\ees
which converges to zero as $t \rightarrow + \infty$. Enlarge $t_3 > 0 $ so that the RHS is less than $\epsilon$ for $t\geq t_3$ to get (\ref{variationslopes}).

%%%ELIMINABILE Titoletto
\subsubsection{Equidistribution on the base and asymptotic of curves.}\label{asymptoticscurvesandequidistributionsec}
Both the equidistribution on the base and the exact asymptotic for the number of curves follow by proving uniform convergence on a large set for the Birkhoff sums of $\chi$ 
%_{[b_1,b_2-\delta[ }$ (characteristic function, where $[b_1,b_2]$ is the base of $R$)$
and $f$ respectively. More precisely, one seeks uniform control for the points of the form $T^{r(a)}b$ where $b$ are the endpoints of the partition intervals $[a,b[$. %First refining the partition further and controlling the measure of the pull-back by $R^{t}$ allow us to control points of the form $T^{r(x,t)}x$ in each partition interval; from those one can control also $T^{r(a)}b$. 

%To guarantee uniform convergence for these points, we refine the partitions further by controlling the pull-back by $R^{t}$ using Lemma \ref{Kochlemma}. The construction has to take into account that this allows to control only points of the form $T^{r(x,t)}x$.

%\paragraph{Reduction of asymptotics to uniform convergence.}
\paragraph{Equidistribution on the base.}
%For brevity, let us write $\chi$ for $\chi_{[b_1,b_2-\delta[ }$.
Let $\widetilde{\eta}_5(t)$ be a narrowing of $\eta_5(t)$ obtained keeping only the central third of each interval:
\bes
\widetilde{\eta}_5(t) \doteqdot  \left\{ \left. \left[ a + (b-a)/3 %\frac{(b-a)}{3}
 , b - (b-a)/3  % \frac{(b-a)}{3}
 \right[ \, \right|\quad  [a,b[ \in \eta_5(t)   \right\} .
\ees

For each $\varepsilon>0$ and $\delta_1>0$, by ergodicity of $T$ and $T^{-1}$ one can find $U_5$ and $N>0$ such that $Leb(U_5)< \delta$ and for each $x\notin U_5$ and $n\geq N$, 
\be \label{unifconvBS}
\left| \frac{\BS{f, T^i}{n}(x)}{n} -1  \right| < \varepsilon ; \qquad  \left| \frac{\BS{\chi, T^{i}}{n}(x)}{n} -(b_2-b_1-2\delta)  \right| < \varepsilon; \qquad i=0,-1. 
\ee
%\begin{eqnarray}
%\left| \frac{1}{n}\BS{f, T}{n}(x) -1  \right| < \varepsilon ; & & \left| \frac{1}{n}\BS{f, T^{-1}}{n}(x) -1  \right| < \varepsilon ; 
%\label{unifconvBS} \\
% \left| \frac{\BS{\chi , T}{n}(x)}{n} -(b_2-b_1-2\delta)  \right| < \varepsilon ;  &&  \left| \frac{\BS{\chi, T^{-1}}{n}(x)}{n} -(b_2-b_1-2\delta)  \right| < \varepsilon . 
%\label{unifconvBS}
%\end{eqnarray}
If $\delta_1$ is given by Corollary \ref{Kochlemmacor} in correspondence of $\delta/3>0$, we get $Leb(R^{-t}(U_5))<\delta/3$.  
Define $\eta_6(t)\subset \eta_5(t)$ by throwing away all intervals $I \in \eta_5(t) $ such that the corresponding $\widetilde{I}$ is completely contained in $R^{-t}(U_5)$. Hence,
%$Leb(\eta_6(t))\geq Leb{\eta_5(t)}-3\delta$, i.e. $Leb(\eta_6(t)) \geq 1-5\delta$,
%\bes
$Leb(\eta_6(t))\geq Leb(\eta_5(t))-\delta  \geq Leb(\eta_{s}(t))-2\delta $ by Lemma \ref{curvesdistsinglemma}.
%\ees

%%%
By construction, for each $I\in \eta_6(t)$, there exists $\overline{x}$ such that $|\overline{x}-a|$, $|\overline{x}-b|> (b-a)/3$ and $T^{r(\overline{x},t)}\overline{x} \notin U_5$ and hence satisfies (\ref{unifconvBS}).
Arguing as in Corollary \ref{growthBSpartitioncor} to prove (\ref{growthDeltaf}), both 
$\Delta f ([a,\overline{x}],t )$ and $\Delta f ([\overline{x},b],t )$, as $t\rightarrow \infty$, are bounded from below by $O ( \log t / \log \log t)$. % and therefore converge to infinity uniformely.
%\bes
%\Delta f ([a,\overline{x}],t ) ,\, \Delta f ([\overline{x},b],t ) \geq  O \left( \log t / \log \log t
%%\frac{\log t }{\log \log t}
% \right) \xrightarrow{t \rightarrow + \infty } + \infty. 
%\ees
As in Lemma \ref{equivalentJlemma}, 
%%%%$r(\overline{x})-r(a) \geq r(T^{r(a)}\overline{x}, \Delta f ([a,\overline{x}],t )$ and $ r(b)-r(\overline{x}) \geq  r(T^{r(\overline{x})}b ,\Delta f ([\overline{x},b],t ))$
\bes
r(\overline{x})-r(a) \geq r(T^{r(a)}\overline{x}, \Delta f ([a,\overline{x}],t ); \qquad r(b)-r(\overline{x}) \geq  r(T^{r(\overline{x})}b ,\Delta f ([\overline{x},b],t )).
\ees
Hence, by the same proof in Lemma \ref{roughlowerboundJlemma}, both $r(\overline{x})-r(a)$ and $r(b)-r(\overline{x})$ tend to infinity uniformely as $t$ increases.
%\be \label{rxrarbratendstoinfty}
%r(x)-r(a), \, r(b)-r(a)   \xrightarrow{t \rightarrow + \infty } + \infty. 
%\ee
Enlarge $t_3$ so that for $t\geq t_3$, both
$r(\overline{x})-r(a) > N$ and $r(b)-r(\overline{x}) > N$.
%\bes
%r(\overline{x})-r(a) > N  , \quad r(b)-r(\overline{x}) > N . 
%\ees
Hence, the estimates in (\ref{unifconvBS}) hold when $n=r(\overline{x})-r(a)$ or $r(b)-r(\overline{x})$ and $x=T^{r(\overline{x},t)} \overline{x}$. 
%\begin{eqnarray}
% \left| \frac{\BS{f,T}{r(b)-r(\overline{x})}(T^{r(\overline{x},t)}\overline{x}) } {r(b)-r(\overline{x})}  -1  \right| &<& \varepsilon ; 
%\qquad   \left| \frac {\BS{f,T^{-1}}{r(\overline{x})-r(a)}(T^{r(\overline{x},t)-1}\overline{x}) }{{r(\overline{x})-r(a)}} -1  \right| < 2\varepsilon ;   \label{unifconvBSfxbar} \\ 
% \left| \frac{\BS{\chi,T}{r(b)-r(\overline{x})}(T^{r(\overline{x},t)}\overline{x})}{r(b)-r(\overline{x})} -(b_2-b_1-2\delta)  \right| &<& \varepsilon ;    \qquad \left| \frac{\BS{\chi, T^{-1}}{r(\overline{x})-r(a)}(T^{r(\overline{x},t)}T\overline{x})}{r(\overline{x})-r(a)} -(b_2-b_1-2\delta)  \right| < 2 \varepsilon . 
%\label{unifconvBSchixbar} 
%\end{eqnarray}
Moreover they also hold for $i=-1$ and $x=T^{r(\overline{x},t)-1} \overline{x}$. To see it, in the case of $f$, use that
%The second estimates in (\ref{unifconvBSfxbar}) can be obtained from the convergence of $\BS{f ,T^{-1}}{r(\overline{x})-r(a)+1}(T^{r(\overline{x},t)} \overline{x}) $ by remarking that
\bes
 \BS{f,T^{-1}}{r(\overline{x})-r(a)}(T^{r(\overline{x},t)-1}\overline{x}) = \BS{f ,T^{-1}}{r(\overline{x})-r(a)+1}(T^{r(\overline{x},t)} \overline{x}) - f(T^{r(\overline{x},t)} \overline{x}) 
\ees
and from Lemma \ref{curvesdistsinglemma} and the analogous of Lemma \ref{roughlowerboundJlemma} for $ \Delta f ([a,\overline{x}],t )$,
%\bes
%\left| \right| \leq  
%\ees
\bes \frac{f(T^{r(\overline{x},t)}\overline{x}) }{r(\overline{x})-r(a) } \leq O\left( \log (\log t)^2 \frac{(\log \log t)^2}{\log t}\right) 
\ees
which can be made arbitrarly small enlarging $t_3$ if necessary.
In the case of $\chi$, just use that $\chi\leq 1$ and $r(\overline{x})-r(a)$ tends to infinity.
%The second estimates (\ref{unifconvBSchixbar}) follows in an analogous way, simply using that $\chi\leq 1$ and $r(\overline{x})-r(a)$ tends to infinity.

%\paragraph{Equidistribution on the base.}
Let us combine these estimates decomposing the Birkhoff sums as
\bes %\label{BSdecompos}
\BS{\chi,T}{r(b)-r(a)} (T^{r(a)} \overline{x} ) 
= \BS{\chi,T^{-1}}{r(\overline{x})-r(a)} (T^{r(\overline{x})-1} \overline{x} ) + \BS{\chi,T}{r(b)-r(\overline{x})} (T^{r(\overline{x})} \overline{x} )
\ees
%Combining (\ref{unifconvBSfxbar}) and (\ref{unifconvBSchixbar}) 
and using that $\frac{r(b)-r(\overline{x})}{r(b)-r(a)} + \frac{r(\overline{x})-r(a)}{r(b)-r(a)}  =1 $. We get
\be\label{BSdecompcons} %\begin{split}&
 \left|  \frac{\BS{\chi,T}{r(b)-r(a)}(T^{r(a)}\overline{x})}{r(b)-r(a)} -(b_2-b_1-2 \delta)  \right|\leq 2\epsilon ,
\ee
%\\ &\leq 
%\\ &
%\frac{r(b)-r(\overline{x})}{r(b)-r(a)}  \left| \frac{\BS{\chi}{r(b)-r(\overline{x})}(T^{r(\overline{x},t)}\overline{x})}{r(b)-r(\overline{x})} - (b_2-b_1-2\delta) \right| + 
%\\& +
% \frac{r(\overline{x})-r(a)}{r(b)-r(a)} \left| \frac{\BS{\chi,T^{-1}}{-(r(\overline{x})-r(a))}(T^{r(\overline{x},t)-1}\overline{x})}{r(\overline{x})-r(a)} -(b_2-b_1-2\delta) \right| < 4\varepsilon.
%    \end{split}
which proves equidistribution on the base (\ref{equidistributionbase}) for $\overline{x}\in I$.

%\paragraph{Reduction of asymptotics to uniform convergence.}
\paragraph{Asymptotic number of curves.}\label{asymptoticscurvessec}
%Let us show that the control on Birkhoff sums for $T^{r(\overline{x},t)}\overline{x}$  given by (\ref{unifconvBSfxbar}) 
%allows to control also $T^{r(a)}b$ and prove:
\begin{lemma}\label{convergenceBSrab} Enlarging $t_3$ if necessary, for each $[a,b[ \, \in \eta_6(t)$, $t\geq t_3$,
\bes \label{BSforb}
\left|  \frac{1}{r(b)-r(a)}  \BS{f}{r(b)-r(a)} (T^{r(a)} b)  -1  \right| \leq  2 \varepsilon.
\ees
\end{lemma}
\begin{proof}
%Let us derive equations analogous to (\ref{unifconvBSfxbar}) also for $T^{r(\overline{x},t)}b$, by proving first that
%\bes
%\left| \BS{f}{r(b)-r(\overline{x})} (T^{r(\overline{x},t)} \overline{x}) - \BS{f}{r(b)-r(\overline{x})} (T^{r(\overline{x},t)} b) \right|  \xrightarrow{t \rightarrow + \infty } 0 ; %\label{differencefuture} \\ 
%\ees
By mean value, there exists $z \in [\overline{x},b]$ such that% the quantity above  is bounded by
\bes
\begin{split}
\left| \BS{f}{r(b)-r(\overline{x})} (T^{r(\overline{x},t)} \overline{x}) - \BS{f}{r(b)-r(\overline{x})} (T^{r(\overline{x},t)} b) \right| \leq  \left| \BS{f'}{r(b)-r(\overline{x})}(T^{r(\overline{x},t)}z) \right|\cdot \\ \cdot (b-\overline{x}) \leq J(t) \sup_{r(\overline{x})  \leq i\leq r(b)} |f'(T^{i}z) | (b-\overline{x}) \leq 
  \frac{ o(\log t) O((\log t)^2) }{t \log \log t}%\xrightarrow{t \rightarrow + \infty } 0 , 
\end{split}
\ees
%%%ELIMINABILE spiagazione
where we used Corollary \ref{Jupperboundlemma} to bound $J(t)$ , Lemma \ref{curvesdistsinglemma} to bound $|f'(T^{i}z)|$ and Property 2 in Proposition \ref{preliminarypartitionprop} to control the size $(b-a)$. % by Property 2 in Proposition \ref{preliminarypartitionprop}.
%The proof of (\ref{differencepast}) is analogous.
Hence, enlarging $t_3$, from the analogous estimate for $T^{r(\overline{x},t)}\overline{x}$, we get for $t \geq t_3$, % since $r(b)-r(\overline{x})>1$,
\be \label{BSTrxbpast}
%\begin{split} \label{BSTrxbpast}
\left| \frac{1}{r(b)-r(\overline{x})}\BS{f}{r(b)-r(\overline{x})}(T^{r(\overline{x},t)}b) -1  \right| \leq 2 \varepsilon .
% \left| \frac{1}{r(b)-r(\overline{x})}\BS{f}{r(b)-r(\overline{x})}(T^{r(\overline{x},t)}\overline{x}) -1  \right| + \\+ \left| \BS{f}{r(b)-r(\overline{x})} (T^{r(\overline{x},t)} \overline{x}) - \BS{f}{r(b)-r(\overline{x})} (T^{r(\overline{x},t)} b) \right| < 2 \varepsilon .
%  \end{split}
\ee
In a similar way, from the analogous estimate for  $T^{r(\overline{x},t)-1}\overline{x}$, we get
\be \label{BSTrxbfuture} 
\left| \frac{1}{r(a)-r(\overline{x})}\BS{f,T^{-1}}{r(a)-r(\overline{x})}(T^{r(\overline{x},t)-1}b) -1  \right|  < 2 \varepsilon .
\ee
Combining (\ref{BSTrxbpast}) and (\ref{BSTrxbfuture}) and decomposing the Birkhoff sums as %Using the same decomposition used for Birkhoff sums of $\chi$ in (\ref{BSdecompos}),
\bes
\BS{f,T}{r(b)-r(a)} (T^{r(a)} b )=\BS{f,T^{-1}}{r(\overline{x})-r(a)} (T^{r(\overline{x})-1} b ) + \BS{f, T}{r(b)-r(\overline{x})} (T^{r(\overline{x})}b)
\ees
we get the Lemma.
\end{proof}

%Let us now show that the asymptotic of curves (\ref{asymptoticscurves}) can be deduced from the convergence of Birkhoff sums in Lemma \ref{convergenceBSrab}.
\begin{lemma}\label{reductionasymptoticslemma}
Enlarging $t_3$ if necessary, for each $[a,b[\, \in \eta_6(t)$, if $t \geq t_3$,
\bes
\left| \frac{ \BS{f}{r(b)-r(a)} (T^{r(a)}b)}{\Delta f} - 1\right|  \leq 
\varepsilon .
\ees
\end{lemma}
\begin{proof}
Since we can rewrite
%%%ELIMINABILE
\bes
\begin{split}
 \BS{f}{r(b)-r(a)} (T^{r(a)}b) &= \BS{f}{r(b)} (b) -  \BS{f}{r(a)} (b) \pm \BS{f}{r(a)} (a) = 
\\ &= \Delta f + \BS{f}{r(b)} (b) -  \BS{f}{r(a)} (a),
\end{split}
\ees
%$ \BS{f}{r(b)-r(a)} (T^{r(a)}b) = \BS{f}{r(b)} (b) -  \BS{f}{r(a)} (b) \pm \BS{f}{r(a)} (a) =  \Delta f + \BS{f}{r(b)} (b) -  \BS{f}{r(a)} (a)$
from $t - f( T^{r(a)} a ) <  \BS{f}{r(a)} (a) \leq t $ and $t-   f( T^{r(b)} b) <  \BS{f}{r(b)} (b)\leq t $, we get
\bes
%\begin{split}
\left| \frac{ \BS{f}{r(b)-r(a)} (T^{r(a)}b)}{\Delta f} - 1\right| % \leq  \left| \frac{ \BS{f}{r(b)} (b) -  \BS{f}{r(a)} (a)}{\Delta f}  \right| \leq \\ 
\leq \frac{\max \{ f( T^{r(b)} b) , f( T^{r(a)}a) \}}{\Delta f} \leq 
\frac{ O((\log  \log t)^2)}{\log t}\rightarrow 0,
%\xrightarrow{t \rightarrow + \infty} 0.
%\end{split}
\ees
where we used by Lemma \ref{curvesdistsinglemma} and (\ref{growthDeltaf}).
\end{proof}

%Recalling that $J([a,b[,t)-1 =  r(b,t)-r(a,t) $, %but since $J([a,b[,t)$ tends to infinity by Lemma \ref{roughlowerboundJlemma},
%\bes
% \left| \frac{J(t)} { r(b,t)-r(a,t) } -  1 \right| < \epsilon,
%% \frac{1}{r(b)-r(a)} =r\left( T^{r(a)} b  , \Delta f(I,t) \right).
%\ees
%for $[a,b[\in \eta_6(t)$ with $t \gg 1$.
%This last remark, combined with 
Lemma  \ref{convergenceBSrab} and Lemma \ref{reductionasymptoticslemma} give
\bes
\left| \frac{ r(b,t)-r(a,t)}{ \BS{f}{r(b)-r(a)} (T^{r(a)}b)} \frac{ \BS{f}{r(b)-r(a)} (T^{r(a)}b)}{\Delta f} - 1\right| \leq (1+\epsilon)
% \frac{(1+\epsilon)^2}{1-4\epsilon} ,
\ees
for an appropriate choice of $\varepsilon$ and $t\geq t_3$. Recalling that $J([a,b[,t)-1 =  r(b,t)-r(a,t) $, this concludes the proof of 
%%%
 the asymptotic of curves (\ref{asymptoticscurves}).

Setting $\eta_{m}(t)= \eta_6(t)$, this completes the verification that the partitions $\eta_{m}(t)$, for an appropriate choice of $\delta$ and $t\geq t_3$, satisfy  all the Properties listed in Proposition \ref{mainpropertiespartitionprop}. 
%Hence $\eta_{m}(t)$ are the partitions which verify (\ref{mainestimate}) and allow to prove mixing by the mixing criterium in Lemma \ref{mixingcriterium}.





 

\subsection*{Acknowledgments.}
I would like to thank my advisor Prof. Ya. G. Sinai for proposing the problem and constantly guiding me with suggestions, patience and encouragement.
I also would like to thank both K. Khanin and A. Avila for useful discussions. 
Thanks also to Pavel Batchourine for listening to some parts of this work.

%{\it Email addresses:}
%{ulcigrai@math.princeton.edu}

\bibliography{biblioflows}
\bibliographystyle{alpha} 



%{\it Email address:}
%{ulcigrai@math.princeton.edu}
\end{document}



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\newblock {\em Ann. Inst. Fourier, Grenoble}, 46:325--370, 1996.

\bibitem[Zor99]{Zo:how}
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\newblock How do the leaves of a closed 1-form wind around a surface?
\newblock {\em Tanslations of American Mathematical Society}, 197:135--178,
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\end{thebibliography}
