On the Unstable Directions and Lyapunov Exponents of Anosov Endomorphisms
F. Micena
A. Tahzibi
Despite the invertible setting, Anosov endomorphisms may have infinitely many unstable directions. Here we prove, under transitivity assumption, that an Anosov endomorphism on a closed manifold $M,$ is either special (that is, every $x \in M$ has only one unstable direction) or for a typical point in $M$ there are infinitely many unstable directions. Other result of this work is the semi rigidity of the unstable Lyapunov exponent of a $C^{1+\alpha$ codimension one Anosov endomorphism and $C^1$ close to a linear endomorphism of $\mathbb{T^n$ for $(n \geq 2).$ In the appendix we give a proof for ergodicity of $C^{1+\alpha, \alpha > 0,$ conservative Anosov endomorphism.
1412.0629.tex
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\begin{document}

\thanks{}

\author{F. Micena}
\address{Departamento de Matem\'atica,
  IM-UFAL Macei\'{o}-AL, Brazil.}
\email{fpmicena@gmail.com}

\author{A. Tahzibi}
\address{Departamento de Matem\'atica,
  ICMC-USP S\~{a}o Carlos-SP, Brazil.}
\email{tahzibi@icmc.usp.br}







\renewcommand{\subjclassname}{\textup{2000} Mathematics Subject Classification}

\date{\today}

\setcounter{tocdepth}{2}

\title{On the Unstable Directions and Lyapunov Exponents of  Anosov Endomorphisms}
\maketitle
\begin{abstract}
Despite the invertible setting, Anosov endomorphisms may have infinitely many unstable directions. Here we  prove, under transitivity assumption, that an Anosov endomorphism on a closed manifold $M,$ is either special (that is, every $x \in M$ has only one unstable direction) or for a typical  point in $M$ there are infinitely many unstable directions. Other result of this work is the semi rigidity of the unstable Lyapunov exponent of a $C^{1+\alpha}$ codimension one Anosov endomorphism and $C^1$ close to a linear endomorphism of $\mathbb{T}^n$ for $(n \geq 2).$  In the appendix we give a proof for ergodicity of  $C^{1+\alpha}, \alpha > 0,$ conservative Anosov endomorphism.
\end{abstract}





%----------------------------------------------------------
\section{Introduction}\label{section.preliminaries}
%----------------------------------------------------------


In $1970s,$ the works \cite{PRZ} and \cite{MP}  generalized the notion of Anosov diffeomorphism for non invertible maps, introducing the notion of Anosov endomorphism. We consider $M$ a $C^{\infty}$ closed manifold.

\begin{definition}\cite{PRZ} \label{defprz} Let $f: M \rightarrow M$ be a  $C^1$ local diffeomorphism. We say that $f$ is an Anosov endomorphism if there is constants $C> 0$ and $\lambda > 1,$ such that, for every $(x_n)_{n \in \mathbb{Z}}$ an $f-$orbit there is a splitting

$$T_{x_i} M = E^s_{x_i} \oplus E^u_{x_i}, \forall i \in \mathbb{Z},$$

which is preserved by $Df$ and for all $n > 0 $ we have

$$||Df^n(x_i) \cdot v|| \geq C^{-1} \lambda^n ||v||, \;\mbox{for every}\; v \in E^u_{x_i} \;\mbox{and for any} \; i \in \mathbb{Z}$$
$$||Df^n(x_i) \cdot v|| \leq C\lambda^{-n} ||v||, \;\mbox{for every}\; v \in E^s_{x_i} \;\mbox{and for any} \; i \in \mathbb{Z}.$$

\end{definition}


Anosov endomorphisms  can be defined in an equivalent way (\cite{MP}):

\begin{definition}\cite{MP} \label{defmp} A $C^1$ local diffeomorphism $f: M \rightarrow M$ is said an Anosov endomorphism if $Df$ contracts uniformly a continuous sub-bundle $E^s \subset TM$ into itself, and the action of $Df$ on $TM/E^s$ is uniformly expanding.
\end{definition}

Sakai, in \cite{SA} proved that, in fact, the definitions $\ref{defprz}$ and $\ref{defmp}$ are equivalent.

A contrast between Anosov diffeomorphisms and Anosov endomorphisms is the non-structural stability of the latter. Indeed, $C^1-$close to any linear Anosov endomorphism $A$ of Torus, Przytycki \cite{PRZ} constructed Anosov endomorphism which has infinitely many unstable direction for some orbit and consequently he showed that $A$ is not structurally stable. However, it is curious to observe that the topological entropy is locally constant among Anosov endomorphisms. Indeed, take the lift of Anosov endomorphism to the inverse limit space (see preliminaries for the definition). At the level of inverse limit space, two nearby Anosov endomorphisms are conjugate (\cite{PRZ}, \cite{BerRov}) and lifting to inverse limit space does not change the entropy.

Two endomorphisms (permitting singularities) $f_1, f_2$ are $C^1-$inverse limit conjugated, if there exists a homeomorphism $h : M^{f_1} \rightarrow M^{f_2}$ such that $h \circ \tilde{f_1} = \tilde{f_2} \circ h$ where $\tilde{f_i}$ are the lift of $f_i$ to the orbit space (see preliminaries).



Denote by $p$ the natural projection $p: \overline{M} \rightarrow M,$ where $\overline{M}$ is the universal covering. Note that an unstable direction $E^u_{\overline{f}}(y)$ projects on an unstable direction of $T_x M, x = p(y)$ following the definition $\ref{defprz}, $ that is $Dp(y) \cdot (E^u_{f}(y)) = E^u(\tilde{x}), $ where $\x =  p (\mathcal{O}(y)).$





\begin{proposition}\label{propMP}\cite{MP} $f$ is an Anosov endomorphism of $M,$ if and only if, the lift  $\overline{f}: \overline{M} \rightarrow \overline{M} $  is an Anosov diffeomorphism of $\overline{M},$ the universal cover of $M.$
\end{proposition}


 An advantage to work with the latter definition is that in $\overline{M}$ we can construct invariant foliations $\mathcal{F}^s_{\overline{f}}, \mathcal{F}^u_{\overline{f}}.$



 Given an Anosov endomorphism and $\x= (x_n)_{n \in \mathbb{Z}}$ an $f-$ orbit we denote by $ E^u(\x)$ the unstable bundle subspace of $T_{x_0}(M)$ corresponding to the orbit $(x_n)_{n \in \mathbb{Z}}.$ In  \cite{PRZ} one constructs examples of Anosov endomorphism such that $E^u({\tilde{x}}) \neq E^u (\tilde{y}),$ when $x_0 = y_0,$ but $(x_n)_n \neq (y_n)_n.$ In fact, it is possible that $x_0 \in M$ has uncountable unstable directions, see \cite{PRZ}.
 An Anosov endomorphism for which $E^u(\x)$ just depends on $x_0$ (unique unstable direction for each point) is called special Anosov endomorphism. A linear Anosov endomorphism of torus is an example of special Anosov endomorphism.




A natural question is whether it is possible  to find an example of  (non special)Anosov endomorphism, such that every $x \in M$ has a finite number of unstable directions. It is also interesting to understand the structure of points with infinitely many unstable directions. For transitive Anosov endomorphisms we prove the following dichotomy:
\begin{theorem}\label{teo1} Let $f: M \rightarrow M$ be a transitive Anosov endomorphism, then:
\begin{enumerate}
\item Either  $f$ is an special Anosov endomorphism,
\item Or there exists a residual subset $\mathcal{R} \subset M,$ such that for every $x \in \mathcal{R},$ $x$ has infinitely many unstable directions.

\end{enumerate}

\end{theorem}

Observe that when $M$ is the torus $\mathbb{T}^n, n \geq 2,$ all Anosov endomorphism of $\mathbb{T}^n$ are transitive, see \cite{AH}.



Analysing the unstable Lyapunov exponents of the Anosov endomorphism,  similarly with \cite{MT} we can prove:

\begin{theorem}\label{teo2} Let $A: \mathbb{T}^n \rightarrow \mathbb{T}^n, n \geq 2$ be a linear Anosov endomorphism, with $dim E^u_A = 1.$ Then there is a $C^1$ open set $\mathcal{U},$ containing $A,$ such that for every $f \in \mathcal{U}$ a $C^{1 + \alpha}, \alpha> 0,$  conservative Anosov endomorphism we have $\lambda^u_f(x) \leq \lambda^u(A),$ for $m$ almost everywhere $x \in \mathbb{T}^n,$ where $m$ is the Lebesgue measure of $\mathbb{T}^n.$
\end{theorem}


\begin{remark}
To prove the Theorem \ref{teo2}, the neighbourhood $\mathcal{U}$ is can be chosen very small, such that every $f \in \mathcal{U}$ has its lift conjugated to $A$ in $\mathbb{R}^n.$ Then by this fact, we can consider a priori that also we have $dim E^u_f = 1.$
\end{remark}



\section{General Preliminaries Results.}\label{section.preliminaries}


In this section we present some classical results on the theory of Anosov endomorphism, that will be important for our purposes for the rest of this work.



\subsection{The Limit Inverse Space.} Consider $(X,d) $ a compact metric space and $f: X \rightarrow X$ a continuous map, we define  a new compact metric space called limit inverse space for $f$ or natural extension of $f,$ being:


$$X^f := \left\{(x_n)_{n \in \mathbb{Z}} \in \prod_{i \in \mathbb{Z}} X_i|\;\; X_i = X, \;\forall \; i \in \mathbb{Z} \;\;\mbox{and}\;\; f(x_i) = x_{i+1} \forall i\in \mathbb{Z}  \right\}.$$


In this text  we denote  $X^f$ by $\widetilde{X}.$ Also we go to denote $\x$ being an element $(x_n)_{n \in \mathbb{Z}}$  of $\widetilde{X}.$ We introduce a metric $\widetilde{d}$ in $\widetilde{X}$ as following:

$$\tilde{d}(\tilde{x}, \tilde{y}) = \displaystyle\sum_{i \in \mathbb{Z}}\frac{d(x_i, y_i)}{2^{|i|}}.$$



It is easy to see  that $(\tilde{X}, \tilde{d})$ is a compact metric space.
Consider $\pi: \widetilde{X} \rightarrow X,$ the projection in the zero coordinate, that is, if $\x = (x_n)_{n \in \mathbb{Z}}, $ then $\pi(\x) = x_0.$
One can verify that $\pi$ is continuous.





\begin{definition} We denote a pre-history of $x,$  a sequence of type $\x_{-} = (\ldots, x_{-2}, x_{-1}, x_0 = x),$ such that $f(x_{-i})= x_{-i + 1}, i =1, 2, \ldots.$
\end{definition}



Denote by $X^f_{-}$ or $\widetilde{X}_{-},$ the space of the all pre-histories with $x_0 \in X.$  The space $(\widetilde{X}_{-}, \widetilde{d})$ also is compact and the distance of two pre-histories of the same point $x_0 \in X$ is $\displaystyle\sum_{i=0}^{\infty}\frac{d_M(x_{-i}, y_{-i})}{2^i}.$



In the Anosov endomorphism context, $E^u(\x)$  depends only on $\x_{-},$ and this is why many times in this work we deal only with pre histories.


\subsection{Some Nice Properties of Anosov Endomorphisms.}
The set of $C^1$ Anosov endomorphisms is open like Anosov diffeomorphisms. However, structural stability in the usual sense does not hold for Anosov endomorphisms (See the correct context for structural stability of Anosov endomorphisms in  Berger-Rovella \cite{BerRov}).
\begin{theorem}[Przytycki \cite{PRZ}, Ma\~{n}\'{e}-Pugh \cite{MP}]
Anosov endomorphisms of a manifold $M$ is an open set in the $C^1$ topology.
\end{theorem}









\begin{theorem} \label{prz} \cite{PRZ} Let $f: M \rightarrow M$ be an Anosov endomorphism, then the map

$$\tilde{x} \mapsto E^u (\tilde{x})$$
is continuous.
\end{theorem}





\begin{definition}
Let $f: M \rightarrow M$ be an Anosov Endomorphism, Denote by $\mathcal{E}^u_f(x) := \displaystyle\bigcup_{\tilde{x}: \pi(\tilde{x}) = x} E^u(\tilde{x})$,  union of all unstable direction defined on $x.$
\end{definition}
Considering the definitions \ref{defmp} and \ref{defprz} a natural question arises: What is the relation between $\mathcal{E}^u_f(x)$ and $ \bigcup_{y \in p^{-1}(x)} Dp(E^u_{\overline{f}}(y)) ?$

Observe that $\mathcal{E}^u_f(x)$ is not necessarily $\displaystyle\bigcup_{\pi(y) = x} Dp(y) \cdot ( E^u_{\widetilde{f}}(y)).$ Indeed, the latter is a countable union and the former may be uncountable  (see \cite{PRZ}.)






\begin{proposition}Let $f: M \rightarrow M$ be an Anosov Endomorphism, then

$$ \mathcal{E}^u_f(x)=\overline{\displaystyle\bigcup_{p(y) = x} Dp(y) \cdot( E^u_{\overline{f}}(y))}.$$

\end{proposition}


\begin{proof}

First of all we would like to mention that $ \mathcal{E}^u_f(x)$ is a closed subset of $u-$dimensional grassmanian of $T_xM.$ This is an immediate corollary of theorem \ref{prz}. Clearly $\displaystyle\bigcup_{\pi(y) = x} Dp(y) \cdot( E^u_{\widetilde{f}}(y)) \subset \mathcal{E}^u_f(x). $  So,  $\overline{\displaystyle\bigcup_{\pi(y) = x} Dp(y) \cdot( E^u_{\overline{f}}(y)} )\subseteq \mathcal{E}^u_f(x).$
Now for reciprocal inclusion,
let $E^u(\x)$ be an unstable direction of $x \in M.$ We want to prove that $E \in \overline{\displaystyle\bigcup_{p(y) = x} Dp(y) \cdot( E^u_{\overline{f}}(y))}.$

We claim that given any finite pre-history $(x_{-k}, \ldots , x_{-2}, x_{-1}, x=x_0 ),$ there is a finite piece of $\overline{f}-$orbit  $(y_{-k}, \cdots, \overline{f}^k(y_{-k})),$ which projects on $(x_{-k}, \ldots , x_{-2}, x_{-1}, x ),$ that is $$\pi( \overline{f}^{j}y_{-k}) = x_{-k+j}, j \in \{1, \cdots, k\}.$$

 Indeed, choose any $y_{-k} \in \overline{M},$ such that $p(y_{-k}) = x_{-k}.$ As $p\circ \overline{f} = f \circ p,$ the piece of orbit of $y_{-k}$ by $\overline{f}$ orbit projects on  $(x_{-k}, \ldots , x_{-2}, x_{-1}, x ).$


Now for each $k$ consider  $\mathcal{O}(y_{-k}),$ the full orbit by $\overline{f}$ of $y_{-k}.$ It is clear that $p(\mathcal{O}(y_{-k}))$ converges to $\tilde{x}$ in $M^f.$ Recall that
\begin{equation} \label{kk}
E^u(p(\mathcal{O}(y_{-k}))) = Dp (E^u(\overline{f}^k(y_{-k}))).
\end{equation}

  By continuity argument (theorem \cite{PRZ}) we have
  $$
  E^u(\pi(\mathcal{O}(y_{-k}))) \rightarrow E^u(\tilde{x})
  $$
  and using \ref{kk} we obtain
  $$
  Dp (E^u(\overline{f}^k(y_{-k}))) \rightarrow E^u(\tilde{x})
  $$
  which completes the proof.

\end{proof}



The next lemma is  useful for the rest of this paper.


\begin{lemma}\label{angle} Suppose that $f: M \rightarrow M$ is an Anosov endomorphism, such that  there are two different unstable directions $E^u_1$ and $E^u_2$ at $x,$ then the angle $\angle(Df^n(x)(E^u_1), Df^n(x)(E^u_2) ),$ goes to zero when $n \rightarrow + \infty.$
\end{lemma}


\begin{proof}  In fact, suppose that $dim(E^s) = k, \dim(E^u) = n .$  Suppose that $E^u_1(x) \neq E^u_2(x),$ for  $x \in M.$ Consider $\{v_1, \ldots, v_n\}$ and $\{u_1, \ldots, u_n\}$ basis for $E^u_1(x)$ and $E^u_2(x)$ respectively. Since $E^u_1 (x) \neq E^u_2 (x)$ there is $u_i,$ say $u_1,$ such that $B = \{u_1, v_1, \ldots, v_n\}$ is a linearly independent set.

 Let $E :=  < u_1, v_1, \ldots, v_n >$ with $dim(E) = n+ 1$ be the subspace generated by $B.$ Observe that $dim(E) + dim(E^s) = n + k + 1 > n+k = dim(T_xM).$ This implies that $E \cap E^s$ is non trivial. Let $0 \neq v_s \in E \cap E^s,$ we have


$$v_s =  cu_1 + v, $$


where   $c \neq 0$ and $ v \in E^u_{1} (x).$


Considering the following properties of vectors in stable and unstable bundles:
\begin{itemize}
\item $||Df^n(x) v_s|| \rightarrow 0,$
 $||Df^n(x) u_1|| \rightarrow + \infty,$
\quad \text{and} \quad $||Df^n(x) v ||\rightarrow + \infty$
\end{itemize}
 It come out  that $\angle([Df^n(x)u_1], Df^n(x)E^u_1(x)) \rightarrow 0.$ In fact the same argument shows that $\angle([Df^n(x)u_i], Df^n(x)E^u_1(x)) \rightarrow 0,$ for all $u_i$ not in $E^u_1(x).$ Thus  $$\lim_{n \rightarrow \infty}\angle(Df^n(x)(E^u_1 (x)), Df^n(x)(E^u_2(x)) )= 0.$$

\end{proof}


\section{Proof of the Theorem \ref{teo1}.}


In the course of the proof of the main result we need to analyse the number of unstable directions as a function of $x \in M$ as follows:
 Let $u: M \rightarrow  \mathbb{N} \cup \{\infty\}$ be defined as  $$u(x) := \#( \mathcal{E}_f^u(x)),$$ which assigns to each $x$ the``number" of all possible unstable directions at $T_x{M}$.

A simple and useful remark is the following:
\begin{lemma} \label{nondecreasing}
 $u(x)$ is non-decreasing along the forward orbit of $x.$
\end{lemma}
\begin{proof}
It is enough to use that $f$ is a local diffeomorphisms and $Df(x)$ is injective. However, we emphasize that it is not clear whether $u(x)$ is constant or not along the orbit. This is because, all the pre history of $x$ is included in the prehistory of $f(x).$
\end{proof}
%\begin{lemma}
%Let $f: M \rightarrow M$ be an Anosov endomorphism. Suppose that $u$ is finite, then $x \mapsto u(x)$ is lower  semi continuous.
%\end{lemma}


%\begin{proof}
%Fix a $x \in M,$ suppose that $u(x) =  k,$ and $E^u_1(x), \ldots, E^u_k(x),$ are the unstable directions for $f$ at $x.$ It is possible $k$ pre histories of $x$ 
%
%$$\x_1:=( \ldots, x_{-2}^1,x_{-1}^1, x) $$
%
%$$\x_2:=( \ldots, x_{-2}^2,x_{-1}^2, x) $$
%
%$$\vdots$$
%
%$$\x_k:=( \ldots, x_{-2}^k,x_{-1}^k, x) $$
%
%such that $E^u_i(x) = E^u(\x_i), i =1, \ldots, k.$
%
%
%Now, consider $N$ a very large integer number and $U_x$ a very small neighborhood of $x,$ such that, for every $y \in U_x$ is possible to define pre-histories
%
%$$\widetilde{y}_i : = ( \ldots,  y_{-N},\ldots, y_{-2}^i,y_{-1}^i, y) , i = 1, \ldots, k $$
%
%such that for each $i$ the $N$ first terms of $\widetilde{y}_i$ is obtained by the same inverse branches of the definition of $\x_i.$
%
%By continuity of local inverse branches and since $U_x$ is very small and $N$ is very large
%
%$$d(\x_i, \widetilde{y}_i) < < 1, $$
%
%such that $E^u(\widetilde{y}_i)$ is very close to $E^u(\x_i),$ by theorem $\ref{prz}.$ Then decreasing $U_x$ and increasing $N$ if necessary there is at least $k= u(x)$ unstable directions at $y \in U_x.$ Thus $x \mapsto u(x)$ is lower semi continuous.
%
%
%\end{proof}








\begin{proposition}\label{quase} Let $f: M \rightarrow M$ be a transitive Anosov endomorphism,  then either there is $x \in M$ such that $u(x) = \infty,$ or $u$ is uniformly bounded on $M,$ in fact in this case $f$ is an special Anosov endomorphism.
\end{proposition}



\begin{proof} Suppose that  $u(x) < \infty$ for all $x \in M.$    Define the sets

$$\Lambda_k =\{x \in M\, | \, u(x) \leq k \}.$$

The sets $\Lambda_k$ are closed. Indeed by continuity argument (theorem \ref{prz}) it comes out that $M \setminus \Lambda_k$ is open. Now observe that
$$ M = \displaystyle\bigcup_{k=1}^{+\infty}\Lambda_k, $$
by Baire  Category theorem,  there is $k_0 \geq 1$ such that $int(\Lambda_{k_0}) \neq \emptyset.$

%As $f $ is transitive, there is $x_0 \in int(\Lambda_{k_0}),$ such that $\{x_n :=f^n(x)| n \geq 0\}$ is dense in $M.$
 Now we claim that $$M= \Lambda_{k_0}.$$ To prove the claim, take $x$ arbitrary in $M$ with $l$ unstable directions and $V_x$ a small neighbourhood  of $x$ such that any point in $V_x$ has at least $l$ unstable direction.  Consider a point with dense orbit in $V_x$ and take an iterate of it that belongs to $\Lambda_{k_0}.$
 By lemma \ref{nondecreasing} we conclude that $l \leq k_0$ and this yields $M = \Lambda_{k_0}.$

Finally, we prove that $ M = \Lambda_1$ implying that $f$ is an special Anosov endomorphism. Suppose that, there is $x \in M$ such that $u(x) \geq 2$ and choose $E^u_1(x), E^u_2(x)$ two different unstable directions at $T_x(M).$ Let  $\alpha > 0 $ be the angle between $E^u_1(x)$ and $E^u_2(x).$



 Consider $U_x$ a small neighbourhood of $x,$ such that every $y \in U_x$ has at least two unstable directions, say $E^u_1(y)$ and $E^u_2(y),$ with $\angle(E^u_1(y), E^u_2(y)) > \displaystyle\frac{\alpha}{2}.$

 Let $x_0$ be a point with dense orbit.  Let $n_1$ be a large number satisfying
 \begin{itemize}
 \item $f^{n_1}(x_0) \in U_x, $
 \item $\angle (Df^{n_1}(x_0)\cdot E,Df^{n_1}(x_0)\cdot F ) < \displaystyle\frac{\alpha}{3} ,$ for any $E,F \in \mathcal{E}^u_f(x_0).$
 \end{itemize}
 The choice of $n_1$ is possible thanks to denseness of the forward orbit of $x_0$ and lemma \ref{angle}.
 By definition of $U_x$ the two above properties imply that  either $E^u_1(f^{n_1}(x_0))$ or $E^u_2(f^{n_1}(x_0))$ is not contained in $Df^{n_1}(x_0)\cdot\mathcal{E}^u_f(x_0).$  So, we obtain
  $$u(f^{n_1}(x_0)) > u(x_0) + 1.$$ By repeating this argument, it is possible to obtain an infinite sequence $ f^{n_k}(x_0)$ such that $$u(f^{n_{k+1}}(x_0)) \geq u(f^{n_{k}}(x_0)) + 1,$$ contradicting $M = \Lambda_{k_0}.$



  %$u(y) \geq u(x),$ for every $y \in U_x.$ Consider $x_n = f^n(x_0),$ a dense orbit in $M,$ and $x_{n_k} $ a subsequence converging to $x.$
%
% Since that $x_{n_k}\rightarrow x,$ there is $k_0$ such that $k \geq k_0,$ we have $x_{n_k} \in U_x,$ in particular $u(x_{n_k}) \geq 2,$ and more, there are at least two directions $E^u_{1, k}(x_{n_k}), E^u_{2, k}(x_{n_k}), $ with
%\begin{equation}\label{angle}
% \angle(E^u_{1, k}(x_{n_k}), E^u_{2, k}(x_{n_k}))\rightarrow \angle(E^u_1(x) , E^u_2(x) ) \geq \alpha > 0,
%\end{equation}
% Since $u$ is finite and constant in $\{x_n\}_n,$ for any two direction $E^u_1(x_0), E^u_2(x_0),$ we have that the angle between them goes to zero by action of $Df^n(x).$ Then the subsequence ${n_k}$ above can be take such that:
%
% \begin{enumerate}
% \item The angle between any two different unstable diretions  $Df^{n_{k+1} - n_{k}}(x_{n_k})\cdot E^u_{1,x_{n_k}} $ and  $Df^{n_{k+1} - n_{k}}(x_{n_k})\cdot E^u_{2,x_{n_k}}$ is very small, where $E^u_{1,x_{n_k}}, E^u_{2,x_{n_k}}$ are unstable directions at $x_{n_k}.$
%
%  \item By the item above and the equation   $(\ref{angle}),$ we have that $E^u_{1, k+1}(x_{n_{k+1}})$ and $E^u_{2, k+1}(x_{n_{k+1}})$ can not be simultaneously images by   $Df^{n_{k+1} - n_{k}}(x_{n_k})$ of unstable directions of $x_{n_k},$ then  $u(x_{n_{k+1}}) \geq u(x_{n_k}) + 1,$
%  \end{enumerate}
%
%
%
%  so the item (2) above implis that $u(x_{n_k}) \rightarrow + \infty,$ but it is a contradiction with the lemma $(\ref{quase}).$ Thus $f$ is a Strong Anosov Endomorphism.


%Observe also, in the case $u(x) < + \infty,$ for any $ x \in M,$ we have $u(x) \leq u(f(x)).$  In particular,  $u$ is not decreasing in $\{x_n\}_{n \geq 0}.$ Since $\{x_n\}_n$ is dense select a subsequence $x_{n_1}, x_{n_2}, \ldots$ such that
%
%$$x_{n_i} \in int(\Lambda_{k_0}), \forall i \geq 0,$$ then
%
%
%$$u(x_{n_i}) \leq k_0, \forall i \geq 0,$$
%
%Since, given $j \geq 0,$ there is $n_k \geq j$ such that $x_{n_k} \in int(\Lambda_{k_0}), $ then
%
%$$u(x_j) \leq u(x_{n_{k}}) \leq  k_0.$$
%
%So $u(x_j) \leq  k_0,$ for every $j \geq 0,$ thus $u(x_n) \leq k_0,$ for any $n \geq 0.$
%
%It remains to prove that $u(x) \leq k_0,$ for every $x \in M.$ In fact, given $x \in M,$ consider $U_x,$ an open neighborhood of $x,$ such that $u(y) \geq u(x),$ for every $y \in U_x.$ Since $x_n$ is dense in $M,$ there is $n_0$ such that $x_{n_0} \in U_x,$ in particular $u(x) \leq k_0.$ Thus $\Lambda_{k_0} = M.$
\end{proof}









%\begin{theorem} Let $f: M \rightarrow M$ be a transitive Anosov Endomorphism,  then
%
%
%\begin{enumerate}
%\item Or every $x \in M$ has exactly one unstable direction, in this case $f$ is a Strong Anosov Endomorphism;
%
%
%\item Either there is a residual set $\mathcal{R} \subset M,$ such that for every $x \in \mathcal{R},$ $x$ posses infinitely many unstable directions. In particular, in this case the set $\{x \in M| u(x) < +\infty\}$ is meager.
%
%
%
%\end{enumerate}
%
%\end{theorem}

\subsection{Ending the Proof of Theorem \ref{teo1}}
To finalize the proof of the theorem \ref{teo1} it remains to prove that $u(x) = \infty,$ for a residual set $\mathcal{R} \subset M,$ whenever $f$ is not special Anosov endomorphism.
In fact, suppose that there is $x \in M,$ such that $u(x) = + \infty.$ Given $k > 0,$ fix exactly $k$ different unstable directions at $x,$ and $U^k_x$ a neighbourhood of $x,$ such that $u(y) \geq k,$ for every $y \in U^k_x.$ Now, since $f$ is transitive, the open set  $V^k = \displaystyle\bigcup_{i \geq 0} f^i(U^k_x) $ is dense in $M.$ Finally, consider



$$\mathcal{R} := \bigcap_{k \geq 1} U_k,$$


which is a residual set. By construction, given $x \in \mathcal{R}$ we have  $u(x) \geq k$  for every $k > 1,$ which implies $u(x) = +\infty.$ The completes the  proof of theorem \ref{teo1}.







\section{ Proof of Theorem \ref{teo2}.}




Given $f: \mathbb{T}^n \rightarrow \mathbb{T}^n$ an Anosov endomorphism, by the proposition \ref{propMP}, the lift $\overline{f}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is an Anosov diffeomorphism.

Let $f_*: \mathbb{T}^n \rightarrow \mathbb{T}^n$ be the linearisation of $f$. By linearisation of $f$ we mean the unique linear endomorphism of torus and homotopic to $f.$ By Theorem 8.1.1 in \cite{AH}, the linearisation map is hyperbolic.



Although $\mathbb{R}^n$ is not compact, since $\overline{f}$ preserves $\mathbb{Z}^n,$  the derivatives of $\overline{f}$ are periodic in fundamental compact domains of $\mathbb{T}^n.$ This periodicity allows us to prove, in the $\mathbb{R}^n$ setting, analogous results to Anosov diffeomorphisms in compact case.

\begin{lemma}\label{abscont} Let $f: \mathbb{T}^n \rightarrow \mathbb{T}^n$ be a $C^{1+\alpha}-$ Anosov endomorphism. Then for $\overline{f}: \mathbb{R}^n \rightarrow \mathbb{R}^n$ there exist $\mathcal{F}^u_{\overline{f}}$  and $\mathcal{F}^s_{\overline{f}}$ transversally absolutely continuous foliations tangent to $E^u_{\overline{f}}$ and $E^s_{\overline{f}}$ respectively.
\end{lemma}

\begin{proof}
Similar to the compact case, \cite{HPS}.
\end{proof}


\begin{definition} \label{quasi isometric}
A foliation $W$ of $\mathbb{R}^n$  is quasi-isometric if there exist positive constants $Q$ and $b$
such that for all $x, y$ in a common leaf of W we have
$$d_W(x, y) \leq Q^{-1} || x - y|| + b.$$
Here $d_W$ denotes the riemannian metric on $W$ and $\|x-y\|$ is the euclidean distance.
\end{definition}

\begin{remark}\label{remarkquasi} Observe that if $||x - y||$ is large enough, we can  consider $b = 0, $ in the above definition.
\end{remark}



\begin{lemma}\label{quasi_iso_fol}
Let $A$ be as theorem \ref{teo2}. If   $f$ is an Anosov endomorphism  $C^1-$ sufficiently close to $A,$ then $\mathcal{F}^{s,u }_{\overline{f}}$ are quasi isometric foliations.
\end{lemma}




The proof of this lemma follows directly from a proposition due to Brin, \cite{Br}.


\begin{proposition}\label{brin} Let $W$ be a $k-$dimensional foliation on the $\mathbb{R}^m.$ Suppose that there is a $(m-k)-$dimensional plane $\Delta$ such that $T_x W(x) \cap \Delta  =\{0\},$ such that $\angle (T_x W(x) , \Delta) \geq \beta > 0,$ for every $x \in \mathbb{R}^m.$ Then $W$ is quasi isometric.
\end{proposition}


\begin{proof}[Proof of the lemma]
 Consider $U $  a $C^1-$open set containing $A,$ such that for every $f \in U,$  $\overline{f}$ and $\overline{A}$ are $C^1$ close in the universal cover $\mathbb{R}^n.$

 The $C^1$ neighborhood $U,$ is taken such that

\begin{equation}\label{angulou}
|\angle (E^u_{\overline{f}}(x), E^u_A ) | < \alpha,
\end{equation}

\begin{equation}\label{angulos}
|\angle (E^s_{\overline{f}}(x), E^s_A ) | < \alpha,
\end{equation}

 for any $x \in \mathbb{R}^n$ where $\alpha$ is an small number less than 1/2$\angle(E^u_A, E^s_A).$
For the foliation $\mathcal{F}^u_{\overline{f}}$ take $\Delta := E^s_A,$ and, for the foliation $\mathcal{F}^s_{\overline{f}},$  $\Delta := E^u_A.$
  Applying the proposition \ref{brin} completes the proof.


 \end{proof}



%\begin{remark} The lemma above is the novelty of this section, the rest will follow analogous to argument in [MT].
%\end{remark}



\begin{corollary}[Nice Properties]\label{nice}
 For any Anosov endomorphism $f: \mathbb{T}^n \rightarrow \mathbb{T}^n$ close to its linearisation $A$, the following properties hold in the universal covering:

\begin{enumerate}


\item For each $k \in \mathbb{N}$ and $C > 1$ there is $M$ such that,
$$||x -  y|| > M \Rightarrow \frac{1}{C} \leq \displaystyle\frac{||\overline{f}^kx - \overline{f}^k y|| }{||A^kx - A^ky||} \leq C.$$

\item  $ \displaystyle\lim_{||y - x || \rightarrow +\infty} \frac{y-x}{||y - x ||} = E_A^{\sigma}, \;\;  y \in \mathcal{F}^{\sigma}_{\bar{f}} (x), \sigma  \in \{s,  u\},$
uniformly.
\end{enumerate}
\end{corollary}

\begin{proof}
The proof is in the lines of \cite{H} and we repeat for completeness. Let $K$ be a fundamental domain of $\mathbb{T}^d$ in $\mathbb{R}^d, d \geq 2.$ Restricted to $K $ we have

$$||\overline{f}^k -  A^k|| < +\infty, $$

For $\overline{x} \in \mathbb{R}^d,$ there are $x \in K$ and $\overrightarrow{n} \in \mathbb{Z}^d,$ such that $\overline{x}  = x + \overrightarrow{n}, $ since $f_{\ast} = A,$ we obtain:

$$ ||\overline{f}^k(\overline{x}) - A^k(\overline{x})|| = ||\overline{f}^k(x + \overrightarrow{n}) - A^k(x +\overrightarrow{ n})|| = ||\overline{f}^k(x) +A^k \overrightarrow{n} - A^kx - A^k\overrightarrow{n}  || < +\infty. $$

Now, for every $x, y \in \mathbb{R}^d,$

$$||\overline{f}^k x - \overline{f}^ky|| \leq ||A^k x - A^k y|| + 2||\overline{f}^k - A^k||_0$$

$$||A^k x - A^ky|| \leq ||\overline{f}^k x - \overline{f}^k y|| + 2||\overline{f}^k - A^k||_0,$$

where,

 $$||\overline{f}^k - A^k||_0 = \max_{x \in K}\{||\overline{f}^k(x) - A^k(x)||\}.$$

Since $A$ is non-singular,  if $||x -  y|| \rightarrow + \infty,$ then $||A^kx - A^k y|| \rightarrow + \infty.$

So dividing both expressions by $||A^kx - A^k y|| $ and doing $||x -  y|| \rightarrow + \infty$ we obtain the proof of the first item.


For the second item, we just consider the case of $E^s_A,$ for $E^ u$ just take $A^{-1}$ and $(\overline{f})^{-1}$  and same proof holds.

Let $|\theta^s| = \max\{\,|\theta| \;| \; \theta\; \mbox{is eigenvalue of $A$ and}\; 0 < |\theta| < 1 \}.$  Fix a small $\varepsilon > 0$ and consider $\delta > 0,$ such that $0 < (1+ 2\delta)|\theta^s| < 1.$ If $f$ is sufficiently $C^1-$close to $A,$ then $\overline{f}$  is an Anosov diffeomorphism  on $\mathbb{R}^d$ with contracting constant less than $(1+ \delta)|\theta^s|.$


Using the hyperbolic splitting, there is $k_0 \in \mathbb{N},$ such that if $v \in \mathbb{R}^d,$  $k > k_0$ and

$$||A^k v || < (1 + 2\delta)^k |\theta^s|^k ||v||,$$

then

$$||\pi^u_A(v)|| < \varepsilon ||\pi^s_A(v)||.$$



Consider $k > k_0$ and $M$ sufficiently large, satisfying  the first item  with $C = 2$ and in accordance with remark \ref{remarkquasi}.

Take $y \in \mathcal{F}^s_{\overline{f}}(x)$ and $||x - y|| > M.$ Let $d^s$ to denote the riemannian distance on stable leaves of $\mathcal{F}^s_{\overline{f}},$ using quasi isometry property of the foliation $\mathcal{F}^s_{\overline{f}},$ we get


$$d^s(\overline{f}^k x, \overline{f}^k y) <  ((1 + \delta)|\theta^s|)^k d^s(x,y) \Rightarrow $$
$$ ||\overline{f}^k x - \overline{f}^k y|| <  ((1 + \delta)|\theta^s|)^k (Q^ {-1}|| x - y||) \Rightarrow $$
$$ ||A^k x - A^k y|| < 2 ((1 + \delta)|\theta^s|)^k (Q^ {-1}|| x - y||).$$

Finally,  for large $k$ we have:

$$2Q^{-1} ((1 + \delta)|\theta^s|)^k  \leq ((1 + 2\delta) |\theta^s|)^k,$$

So,

$$||\pi^u_A(x - y)|| < \varepsilon ||\pi^s_A(x- y)||.$$














\end{proof}



\begin{lemma} \cite{MT} \label{linalg}
 Let $f : \mathbb{T}^d \rightarrow \mathbb{T}^d$ be an Anosov endomorphism close to  $A: \mathbb{T}^d \rightarrow \mathbb{T}^d,$ such that $dim E^u_A = 1.$ Then   for all $n \in \mathbb{N}$ and $\varepsilon > 0$ there exists $M$ such that for $x, y$ with $y \in \mathcal{F}^u_{\overline{f}}(x)$ and $||x -  y||> M$ then
 $$
   (1 - \varepsilon)e^{n\lambda^{u}_A } ||y -x|| \leq \|A^n(x) - A^n(y)\| \leq (1 + \varepsilon)e^{n\lambda^{u}_A } ||y -x||
 $$
where $\lambda^{u}$ is the Lyapunov exponent of $A$ corresponding to $E^{u}_A.$
 \end{lemma}

\begin{proof} Denote by $E^{u}_A$ the eigenspace corresponding to $\lambda^{u}_A$ and $|\mu| = e^{\lambda^{u}_A},$ where $\mu$ is the eigenvalue of $A$ in the $E^{u}_A$ direction.

Let  $N \in \mathbb{N}$ and choose $x, y \in \mathcal{F}^{u}_f(x),$ such that $|| x - y || > M.$ By corollary \ref{nice},we have

$$ \frac{x - y}{|| x - y||} = v + e_M,$$

where the vector $v = v_{E^{u}_A}$ is a unitary eigenvector of  $A,$ in the  $E^{u}_A$ direction and  $e_M$ is a correction vector that converges to zero uniformly as  $M$ goes to  infinity. We have

$$A^N \left( \frac{x - y}{|| x - y||} \right) = \mu^N v + A^N e_M = \mu^N \left(\frac{x - y}{|| x - y||} \right) -\mu^N e_M  + A^N e_M  $$

It implies that

\begin{align*} || x - y || (|\mu|^N - |\mu|^N ||e_M|| - ||A||^N || e_M||) \leq   || A^N (x - y)|| \\ \leq || x - y || (|\mu|^N + |\mu|^N ||e_M|| + ||A||^N || e_M||).
\end{align*}

Since $N$ is fixed, we can choose  $M > 0,$ such that

$$ |\mu|^N ||e_M|| + ||A||^N || e_M|| \leq \varepsilon |\mu|^N.$$

and the lemma is proved. \end{proof}






\begin{remark}
By Multiplicative Ergodic Theorem for endomorphisms (\cite{QXZ}) the unstable Lyapunov exponent for a typical point is independent of unstable direction. We denote by $\lambda^u(x) = \lambda^u(\tilde{x})$ the unique unstable Lyapunov exponent of $x$ in our context where $\dim (E^u) =1.$
\end{remark}
%First, note that, fixing a regular point $x \in M,$ the sum of positive Lyapunov exponent spaces $\displaystyle\bigoplus_{\lambda_i > 0} E(\x, \lambda_i),$ has dimension $1,$ for every $\x$ such that $\pi(\x) = x.$


%Fix $F$ a direction provided from Oseledc theorem for endomorphism, such that, $F$ is positive Lyapunov exponent $\lambda_F(\x)$ for $\x,$ with $x_0 = x.$    If $E \neq F$ is a the direction of positive Lyapunov exponent of $x$ or a unstable direction,  then similarly to lemma \ref{angle} there are $v_1 \in E\setminus \{0\}, $ $v_2 \in F\setminus \{0\}$ and $v \in E^s(x)\setminus \{0\},$ such that

%$$v_1 + v = v_2,$$

%Applying $Df^n(x) $ in both sides, we have
%$$Df^n(x) \cdot v_1 +Df^n(x) \cdot v = Df^n(x) \cdot v_2, $$
%since $||Df^n(x) \cdot v||\rightarrow 0$ when $n\rightarrow+\infty,$ taking $``\frac{1}{n} \log"$ in both sides we have
%$$\lim_{n \rightarrow+ \infty} \frac{1}{n} \log||Df^n(x) \cdot v_1|| = \lim_{n \rightarrow+ \infty} \frac{1}{n} \log||Df^n(x) \cdot v_2|| = \lambda_F(\x).$$
%So, we prove the lemma and $\lambda_{+}(x) = \lambda_{F}(\x),$ where $F$ can be an unstable direction of $x$ or a positive Lyapunov direction for a $\x$ with $x_0 = x.$




\begin{theorem}[Theorem \ref{teo2}] Let $A: \mathbb{T}^n \rightarrow \mathbb{T}^n, n \geq 2$ be a conservative linear Anosov endomorphism, with $dim E^u_A = 1.$ Then there is a $C^1$ open set $\mathcal{U},$ containing $A,$ such that for every $f \in \mathcal{U}$ a $C^{1+\alpha}, \alpha > 0$  conservative Anosov endomorphism we have $\lambda^u_f(x) \leq \lambda^u_A,$ for $m$ almost everywhere $x \in \mathbb{T}^n,$ where $m$ is the Lebesgue measure of $\mathbb{T}^n.$
\end{theorem}


\begin{proof}


Suppose by contradiction that there is a  positive set $Z \in \mathbb{T}^n,$ such that, for every $x \in Z$ we have  $\lambda^u_{\overline{f}}(x) > (1 + 5 \varepsilon) \lambda^u_A,$ for a small $\varepsilon > 0.$
Since $\overline{f}$ is  $C^{1+\alpha},$ the unstable foliation $\mathcal{F}_{\overline{f}}^u$ is upper absolutely continuous. So, there is a positive Lebesgue measure set $B \in \mathbb{R}^n$ such that for every point $x \in B$ we have

\begin{equation}
 m^u_x(\mathcal{F}^u_{\overline{f}}(x) \cap Z) > 0
\label{1}
\end{equation}
where $m^u_x$ is the Lebesgue measure of the leaf $\mathcal{F}^u_{\overline{f}}(x)$.
Choose a $p \in B$ satisfying (\ref{1}) and consider an interval $[x,y]_u \subset \mathcal{F}^u_{\overline{f}}(p) $ satisfying
 $m^u_p([x,y]_u \cap Z) > 0$ such that the length of $[x,y]_u$ is bigger than $M$ as required in the lemma \ref{linalg} and corollary \ref{nice}. We can choose $M$ such that

$$||Ax - Ay|| < (1 + \varepsilon)e^{\lambda^u_A } ||y -x|| $$

and

$$\frac{|| \overline{f}(x) - \overline{f}(y)|| }{ ||Ax - Ay||} < 1 + \varepsilon. $$


 whenever
 $d^u(x, y) \geq M,$ where $d^u$ denotes the riemannian distance in unstable leaves. The above equation implies that $$ || \overline{f}(x) -\overline{f}(y)|| < (1+ \varepsilon)^2 e^{\lambda^u_A} || y - x||.$$


Inductively, we assume that for  $n \geq 1$ we have

\begin{equation}
|| \overline{f}^n(x) - \overline{f}^n(y)||< (1+\varepsilon)^{2n} e^{n \lambda^u_A }|| y - x||. \label{induction}
\end{equation}

 Since $f$ expands uniformly  on the $u-$direction we have $d^u(\overline{f}^n(x), \overline{f}^n(y)) > M,$ consequently


\begin{eqnarray*}
||\overline{ f}(\overline{f}^nx) - \overline{f}(\overline{f}^ny)|| &<& (1+\varepsilon)|| A(\overline{f}^nx) - A(\overline{f}^ny)|| \\  &<& (1 + \varepsilon)^2 e^{\lambda^u_A} || \overline{f}^nx - \overline{f}^n y||\\ &<&
 (1+\varepsilon)^{2(n+1)} e^{(n+1)\lambda^u_A}.
\end{eqnarray*}


For each $n > 0,$ let $A_n \subset Z$ be the following set

$$A_n = \{ x \in Z \colon\;\; \|D\overline{f}^k(x)|E^u_{\overline{f}}(x) \| > (1+2\varepsilon)^{2k} e^{k\lambda^u_A} \;\; \mbox{for any} \;\; k \geq n\}. $$
We have $m(Z) > 0$ and $Z_n := (A_n \cap Z) \uparrow Z,$ as $(1 + 5 \varepsilon) > (1 + 2\varepsilon)^2,$ for small $\varepsilon > 0.$


Define the number $\alpha_0 > 0$ such that:

$$\displaystyle\frac{m^u_p([x,y]_u \cap Z)}{m^u_p([x,y]_u)} = 2 \alpha_0.$$

Since $Z_n \cap [x,y]_u \uparrow Z \cap [x,y]_u, $ there is $n_0 \in \mathbb{N} ,$ such that $n \geq n_0,$ then

 $$ m^u_p ([x,y]_u \cap Z_n) = \alpha_n \cdot m^u_p([x,y]_u),$$

for $\alpha_n > \alpha_0.$

Thus, for $n \geq n_0$ we have:

\begin{eqnarray}
||\overline{f}^nx - \overline{f}^ny || &>&  Q \displaystyle\int_{[x,y]_u \cap Z_n} ||Df^n(z)|| dm^u_p(z) >  \\ &>&
 Q (1+ 2\varepsilon)^{2n}
e^{n \lambda_A^u } m^u_p ([x,y]_u \cap Z_n) \\ &>& \alpha_0 Q^2 (1 + 2\varepsilon)^{2n} e^{n\lambda^u_A} \|x-y\|. \label{conclusion}
\end{eqnarray}
The inequalities $(\ref{induction})$ and $(\ref{conclusion})$ give a contradiction.

%We conclude that $\lambda^u(\widetilde{f},x) \leq \lambda^u(A),$ for almost everywhere $x \in \mathbb{R}^2.$
%Considering the inverse $\widetilde{f}^{-1} $ we conclude that
%$ \lambda^s(A) \leq  \lambda^s(f,x)$
%for  almost every $x \in \mathbb{R}^2.$
\end{proof}

%\begin{corollary} Consider $A: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ a Linear Anosov Endomorphism and $f$ a $C^2$ Anosov Endomorphism $C^1-$close to $A.$ Suppose that $\lambda^u_{\widetilde{f}}(x) < \lambda^u_A,$ for $m$ a.e. $x \in \mathbb{T}^2.$ Then the measures with maximal entropy are not absolutely continuous with $m.$
%\end{corollary}
%
%
%\begin{proof} Since $f$ is a Strong Anosov Endomorphism, there is a unique unstable direction defined for every $x \in \mathbb{T}^2.$ So, for a.e. $x \in \mathbb{T}^2$ we have
%
%
%
%
%$$\lambda^u_{\widetilde{f}}(\widetilde{x}) = \lambda^u_{f}(x),$$
%
%
%where $\pi(\widetilde{x})  = x.$
%
%Moreover, since $f$ is $C^1$ close to $A,$ then $\theta^f$ and $\theta^A$ are conjugated, so
%
%
%$$h_{top}(f) = h_{top}(\theta^f) = h_{top}(\theta^A) = h_{top}(A) = \lambda^u_A. $$
%
%
%If $\mu$  is a measure with maximal entropy for $f,$ the it can not be absolutely continuous w.r.t. $m.$ In fact if $\mu$ was absolutely continuous w.r.t. $m,$ by Pesin formula
%
%
%$$\lambda_A = h_{top}(A) = h_{top}(f) = h_{\mu } = \int_{\mathbb{T}^2} \lambda^u_f(x) d\mu < \lambda^u_A.  $$
%
%Absurd.
%
%
%
%
%\end{proof}
%
%
%
%
%
%
%\begin{theorem} Let $f: \mathbb{T}^2 \rightarrow \mathbb{T}^2$ be a $C^2-$Anosov endomorphism on $\mathbb{T}^2.$ If $\mathcal{F}^s_{\widetilde{f}}$ and $\mathcal{F}^u_{\widetilde{f}}$ are quasi isometric foliations, then $f_{\ast}:= A$  induce a Linear Anosov Endomorphism of $\mathbb{T}^2$ and  the set $\{x| \; \lambda^u_{{f}} \;(x) > \lambda^u_A\}$ has zero area.
%\end{theorem}
%
%
%\begin{proof}
% Firs we prove that if $\mathcal{F}^s_{\widetilde{f}}$ and $\mathcal{F}^u_{\widetilde{f}}$ are quasi isometric foliations, then $A$ is hyperbolic.
%
%For  $x \in \mathbb{R}^2$ consider  $\mathcal{F}^u_{\widetilde{f}}(x).$ Take a sequence $y_n \in \mathcal{F}^u_{\widetilde{f}}(x),  $ such that
%
%$$||x - y_n || \rightarrow +\infty $$
%
%and
%
%
%$$\frac{x- y_n}{||x -y_n||} \rightarrow v \in S^1.$$
%
%
%Fix $k > 1$ a large number. For large $n's,$ we use the firt part of lemma $\ref{nice},$ and of by quasi isometry of $\mathcal{F}^u_{\widetilde{f}}$ we obtain
%
%
%$$||A^k (x - y_n)|| \geq ||f^k x- f^ky_n|| \geq C \mu^k ||x - y_n|| > \lambda^k||x - y_n||  \Rightarrow A^k\left(\frac{x - y_n}{||x - y_n||}\right) > \overline{\mu}^k,$$
%
%for some  $\overline{\mu} > 1,$ then doing $n \rightarrow +\infty$ we have $||A^k v|| > 1. $ Then $A^k$ can not have two eigenvalues in $\{z \in \mathbb{C} | |z| \leq 1\}. $  Analogously, in the stable direction we conclude that $A^k$ has eigenvalues with norm bigger than 1. Thus $A$ is hyperbolic (diagonalizable).
%
%
%Moreover, taking $k$ increasing we have that $|\theta^u| \geq \mu$ and analogously $|\theta^s| \leq \lambda,$ where $\theta^u$ and $\theta^s$ are the eigenvalues of $A$ and $\mu, \lambda$ are the constants of the definition of Anosov Endomorphism for $f.$ In particular $f$ and $A$ satisfies lemma \ref{nice}.
%
%The rest of the proof, follows from   lemma \ref{nice} and after the same proof of theorem \ref{semirig}.
%
%
%\end{proof}
%


\section{Appendix: Ergodicity Of Anosov Endomorphisms }


In this section we establish the ergodicity of $C^{1+\alpha}$ conservative Anosov Endomorphism. Before this, we remember a classical  result, see \cite{QXZ}.

\begin{proposition} Let $T:  X \rightarrow X $ a continuous map of a compact metric space. For any $T-$invariant Borel probability
measure $\mu$ on $X,$ there exists a unique $\tilde{T}-$ invariant Borel probability measure
$\tilde{\mu}$ on $X^T$ such that $\pi_{\ast}\tilde{\mu} = \mu.$ In particular for a measurable set $A \subset X,$ we have $\mu(A) = \tilde{\mu}(\pi^{-1}(A)). $ Moreover $\mu$ is ergodic if, and only if, $\tilde{\mu}$ is ergodic. Here $\pi: X^T \rightarrow X$ is the projection in the zero coordinate.
\end{proposition}


To prove the ergodicity, we use the theory of SRB-measures of endomorphisms. We suggest the reader to see \cite{QZ}, for further details on SRB theory of  endomorphisms. See also \cite{BT} for the number of ergodic SRB measures for surface endomorphisms in terms of homoclinic equivalence classes.


\begin{theorem}
 Let $f: M \rightarrow M$ be a  $C^{1+\alpha}-$ conservative Anosov Endomorphism. Then $(f,m)$ is ergodic, where $m$ is a volume form on $M.$
\end{theorem}


The above theorem seems to be folklore in the ergodic theory of hyperbolic dynamics. However,  we did not find any written proof.
Initially, observe that if $f$ is an Anosov endomorphism, then $f$ is Axiom. $A$ . Since we are supposing that $f$ is $m-$preserving, then $\Omega(f) = M$ and $f$ is transitive (see \cite{MP} or \cite{PRZ}).



\begin{proof} By Pesin formula for endomorphisms, we know that

$$h_m(f) = \int_M \sum_{i} \lambda_i^{+}(x)m_i(x) dm,$$


where $m_i(x)$ is the algebraic multiplicity of $\lambda_i(x).$ In fact we want to prove that the ergodic decomposition of $m$ is trivial. If $\mu$ is not ergodic, by ergodic decomposition theorem we can suppose that $m$ admits at least two ergodic components $\mu_1$ and $\mu_2,$  such that

\begin{equation}\label{SRB}
h_{\mu_k}(f) = \int_M \sum_{i} \lambda_i^{+}(x)m_i(x) d\mu_k, k=1,2.
\end{equation}
Denote by $B_i = B(\mu_i), i=1,2$ the basins of measures $\mu_1$ and $\mu_2$ respectively,

$$ B_i = \left\{ x \in M| \; \frac{1}{n} \sum_{j = 1}^{n-1} \varphi(f^j(x)) \rightarrow \int_M \varphi d\mu_i\right\}, i =1,2,$$

for every $\varphi \in C^0(M),$ we have $\mu_i(B_i) = 1.$


By the SRB characterization given in \cite{QZ}, the measures $\mu_1$ and $\mu_2$ are SRB, since $\mu_1$ and $\mu_2$ satisfies the formula $(\ref{SRB}).$

Moreover, using the SRB theory developed in \cite{QZ}, the measures $\tilde{\mu}_1, \tilde{\mu}_2,$ are SRB measure, and for $\tilde{m},$ a.e. $\tilde{x} \in M^f,$ we have

$$\pi(\tilde{\mu}_i^{\eta(\tilde{x})}(\tilde{x})) << m^u_{\tilde{x}}, i =1,2,$$

where $\eta(\tilde{x})$ is the atom of a subordinated partition for $\tilde{m}$ and $m^u_{\tilde{x}},$ is the Lebesgue volume defined on $\mathcal{F}^u_f({\tilde{x}}).$

Since $\tilde{\mu}_i(B(\tilde{\mu}_i)) = 1, i =1,2$ we have that $\mu_i(\pi(B(\tilde{\mu}_i))= 1.$ By absolute continuity of conditional measures with respect to Lebesgue measure (in fact equivalence), there exist  $\tilde{x}_1, \tilde{x}_2,$ such that the set

$$F^u_i := B_i \cap \pi(B(\tilde{\mu}_i))\cap \pi(\eta(\tilde{x}_i)), i = 1, 2 $$

has full Lebesgue measure in $\pi(\eta(\tilde{x}_i)), i =1,2.$

 Now we saturate $F^u_i $ by leaves of $\mathcal{F}^s_{f}.$ That is we take $D_i: = \displaystyle\bigcup_{z \in F^u_i } \mathcal{F}^s_f(z),$ as union of stable leaves through points of $F^u_i.$
As $\mathcal{F}^s_f$ is a  continuous foliation,  these saturations contain open sets modulus zero (w.r.t. $m$).

Now, if $z_i \in D_i,  $ then $\mathcal{F}^s_f(z_i )$ intersects $F^u_i$ in a point $y_i.$ Since $y_i, z_i$ are in the same stable leaf

$$ \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{j = 1}^n \varphi(f^j(z_i)) =  \lim_{n \rightarrow \infty}  \frac{1}{n} \sum_{j = 1}^n \varphi(f^j(y_i)) \rightarrow \int_M \varphi d\mu_i , i =1,2 $$


thus $D_i \subset B_i, i =1,2.$




Now, since $f$ is transitive, there is $N$ such that $f^N(D_1) \cap D_2 \neq \emptyset$ in an open set (modulo a null $m$ set), since $f^N(B_1) \subset B_1,$ in particular $B_1 \cap B_2 \neq \varnothing,$ so $\mu_1 = \mu_2,$ absurd.

\end{proof}


%
%\begin{remark} In fact the proof above does not neglect the case $f: M \rightarrow M$ be a $C^{1+\alpha}$ conservative Anosov diffeomorphism. We have other proof of the ergodicity in the diffeomorphism setting, which does not argue with Hopf method.
%\end{remark}


















%
%
%
%
%
%
%
%\begin{lemma}\label{abscont} Let $f: M \rightarrow M$ be a $C^2-$ Strong Anosov Endomorphism. Then the foliations $\mathcal{F}^u_f$ and $\mathcal{F}^u_f$ are transversally absolutely continuous.
%\end{lemma}
%
%\begin{proof}
%Similar to compact case(ESPERAMOS QUE SIM).
%\end{proof}
%
%
%
%
%
%\begin{theorem} [Birkhoff Ergodic Theorem in $M^f$] Let $T: (X,d) \rightarrow (X,d)$ a measurable function, preserving $\mu, $ a borelian measure on a compact metric space $(X,d).$ Let $\varphi: X \rightarrow \mathbb{R},$ a $\mathcal{L}^1(X)$ map. Define $\varphi^f: X^f \rightarrow \mathbb{R}, \varphi^f(\overline{x}) = \varphi(x_0), $ a $\mathcal{L}^1(X^f).$ Then for $\widetilde{\mu}$ a.e. $\overline{x} \in X^f$ we have
%
%$$ \lim_{n \rightarrow + \infty} \frac{1}{n} \sum_{i = 1}^n \varphi(x_i) = \lim_{n \rightarrow + \infty} \frac{1}{n} \sum_{i = 1}^n \varphi(x_{-i}). $$
%
%\end{theorem}
%
%
%
%\begin{lemma} Let $f: M \rightarrow M,$ be a conservative and decomposable Anosov endomorphism, with pieces $P_i, i = 1, \ldots, d,$ where $d  = deg(f) > 1.$ Consider $U = \displaystyle\bigcup_{i = 1}^d P_i, $ then $m\left(\displaystyle\bigcap_{n \in \mathbb{Z}}f^n(U) \right) = 1, $ and for every $x \in \displaystyle\bigcap_{n \in \mathbb{Z}}f^n(U), $ $x$ has exactly one pre image per piece.
%\end{lemma}
%
%
%\begin{proof}
%Since $f$ is local diffeomorphism, for every $N$ a null volume set, we have $m(f(N)) = 0, $ furthermore since $f$ is conservative $m(f^{-1}(N) ) = 0,$ follows that $m(f^n (U)) = 1,$ for any $n \in \mathbb{Z},$ so
%
%$$m\left(\bigcap_{n \in \mathbb{Z}}f^n(U) \right) = 1.$$
%
%
%In fact, the set  $T = \displaystyle\bigcap_{n \in \mathbb{Z}}f^n(U), $ is the set of the points  $x \in M,$ such that forward orbit of $x$ is in $U,$ and moreover, any pre-history of $x$ is in $U.$ If $x \in T,$ $f^{-1}(\{x\}),$ has $d$ pre images in $U,$ note that there is not two pre image in the same pice $P_i,$ since $f|P_i$ is injective.
%
%
%\end{proof}
%
%
%
%
%As the previous lemma, we can define $d$ inverses $f_i: (f(P_i)) \rightarrow P_i,$ such that $f(f_i(x)) = x.$ Of course, by inverse function theorem $f_i$ are $C^r,$ if $f$ is $C^r.$
%Now, given a pre-history $(\ldots, x_{-2}, x_{-1}, x_0 = x),$ for an $x \in T,$ we can associate a sequence $\overline{a} \in \{1, \ldots, d\}^{\mathbb{N}},$  $\overline{a}= (i_1, i_2, i_3, \ldots),$ if, and only if,
%
%$$ x_{-1} = f_{i_1}(x_0)$$
%
%$$ x_{-2} = f_{i_2}(x_1)$$
%
%$$ \vdots$$
%
%$$ x_{-n} = f_{i_n}(x_{-n + 1}), n \geq 1.$$
%
%
%
%\begin{lemma} In the hypothesis of the previous lemma, for each $x \in T,$ there is a bijection with pre-histories of $x$ and sequences in $\{1, \ldots, d\}^{\mathbb{N}},$ defined like above.
%\end{lemma}
%
%
%\begin{definition} Let $x \in T,$ like hypothesis above the unique sequence in $\{1, \ldots, d\}^{\mathbb{N}},$ associated to a pre-history $x^{-} = (\ldots, x_{-2}, x_{-1}, x_0 = x),$ we call that Code of  the pre-history $x^{-}.$ Also, given $\overline{a} \in \{1, \ldots, d\}^{\mathbb{N}},$ we denote by $\overline{a}(x)$ the pre-history of $x,$ defined by $\overline{a}.$
%\end{definition}
%
%
%
%
%\begin{lemma} Let $f: M \rightarrow M,$ be  a $C^2$ conservative and decomposable Anosov Endomorphism, $deg(f) = d > 1.$ Consider $\varphi: M \rightarrow \mathbb{R}  $ a $m$ measurable observable. Then there is $\overline{a} \in \{1, \ldots, d\}^{\mathbb{N}},$ such that for $m$ a.e. $x \in M,$ the pre-history $\overline{a}(x)$ satisfies
%
%$$ \lim_{n \rightarrow + \infty} \frac{1}{n} \sum_{i = 1}^n \varphi(f^i(x)) = \lim_{n \rightarrow + \infty} \frac{1}{n} \sum_{i = 1}^n \varphi(x_{-i}). $$
%\end{lemma}
%
%
%\begin{proof} Let $C \subset M^f$ be the set of holds the Birkhoff Ergodic Theorem in $M^f.$ First we note that $m(p(C)) = 1,$ in fact
%
%
%$$m(p(C)) = \widetilde{m}(p^{-1}(p(C))) \geq \widetilde{m}(C) = 1.$$
%
%
%Now given $N \geq 1,$ consider all of $d^N$ finite sequences $(i_i, \ldots, i_N) \in \{1, \ldots, d\}^N.$ For each sequence $s \in \{1, \ldots, d\}^N ,$ consider for each $x \in T$ the pre-histories of $x$ such that the $N$ first terms is defined by $s,$ it means, if $s = (i_i, \ldots, i_N), $ then
%
%
%
%
%$$x_{-1} = f_{i_1}(x)$$
%
%$$x_{-2} = f_{i_2}(x_{-1})$$
%
%$$\vdots$$
%
%$$x_{-N} = f_{i_N}(x_{-N + 1})$$
%
%the other terms is free.
%
%For each $s \in \{1, \ldots, d\}^N ,$ denote by $C_N(s) $ all the sequences like above. We have that
%
%$$p^{-1}(T) =\displaystyle\bigcup_{s \in  \{1, \ldots, d\}^{{N}}} C_N(s), $$
%
%moreover, since  $m(T) = 1 \Leftrightarrow \widetilde{m}(p^{-1}(T)) = 1$ so, at least one of the  $C_N(s) $ has $\widetilde{m}$ positive measure, for some $s_0 \in \{1, \ldots, d\}^{N}.$
%
%Now we decompose the $C_N(s_0) $ above in unions of sets  $C_{N+ 1}(s_1),$ where $s_1$ denotes a sequence in $ \{1, \ldots, d\}^{N+1},$ witch the $N$ first terms coincides with $s.$ There is a $s_1,$ such that $C_{N+ 1}(s_1) \subset C_{N}(s_0) $ with positive measure $\widetilde{m}.$ Proceeding this argument we obtain a infinite sequence of sets
%
%$$C_{N}(s_0) \supset C_{N + 1}(s_1) \supset C_{N + 2}(s_2) \supset \ldots, $$
%
%
%all with positive $\widetilde{m}$ measure. For simplify, we denote the set above simply by $C_n = C_{N + n}(s_n), n \geq 0.$
%
%
%We affirm that $m(p(C_0 \cap C)) =  1.$ In fact, suppose that there is  $D \subset M$ such that $p(C_0 \cap C) \subset M\setminus D, $ with $m(D) > 0.$ Since $f_i$ are local diffeos, they sends $m$ positive measure set in $m$ positive measure sets. In particular
%
%$$m(f_{i_N} \circ \ldots f_{i_2} \circ f_{i_1} (D\cap T)) > 0$$
%
%since $m(p(C)) = 1,$ we have
%
%$$m((f_{i_N} \circ \ldots f_{i_2} \circ f_{i_1}) (D\cap T) \cap p(C)) > 0,$$
%
%it means that if  $x \in f_{i_N} \circ \ldots f_{i_2} \circ f_{i_1} (D \cap T) \cap p(C), $ then there is  a sequence $\overline{a} \in \{1, \ldots, d\}^{\mathbb{N}},$ such that the past Birkhoff average following $\overline{a}$ coincides with the future Birkhoff average of $x.$ Finally the same holds for $y = f^N(x) \in D\cap T.$
%
%In particular the past  and future Birkhoff averages of $y$ coincides, since for the past we consider the pre-history defined by $\overline{b},$ such that $\overline{b}_k = s_k = i_k, $ where $i_k$ is the $k-$esimo term of $s_0,$ for $1 \leq k \leq N$ and $\overline{b}_{N + k} = \overline{a}_k,$ for every $k \geq 1.$ It means that $p(C_0 \cap C) \cap D \neq \emptyset,$ that is absurd. For the other $C_k = C_{N + k},$ the proof is merely change of the notation.
%
%
%Now consider $B = T \cap\displaystyle\bigcap_{k \geq 0} p(C_{N + k}\cap C),$ we have $m(B) = 1.$  Take a infinite sequence $S_{\infty} \in \{1, \ldots d\}^{\mathbb{N}},$ with the property
%
%$$S_{\infty} \; \mbox{coincides with $s_{n}$ until at least the $N+n$ terms, for all $n \geq 1$ }. $$
%
%
%\begin{center}
%\textit{To finish, $B$ is the set of $x \in T,$ such that  the past Birkhoff average coincides with future Birkhoff average, considering some pre-history of $x$ defined by some $\overline{s} \in \{1,\ldots, d\}^{\mathbb{N}},$ that  coincides with $s_n,$ for the first $N + n$ terms, for every $n,$ }
%\end{center}
%
%translating:
%
%For every $x \in B,$  the past Birkhoff average coincides with future Birkhoff average, considering the pre-history of $x$ defined by $S_{\infty}.$
%\end{proof}
%
%\subsection{Proof of the Ergodicity: The Anosov-Hopf Argument Adapted }
%
%Let $\overline{a} = S_{\infty},$ $\varphi$ (a continuous observable) and  $B$ like previous lemma.
%
%
%
%
%
%
%
%For $x \in M$ consider $\mathcal{U}_x$ a small neighborhood of $x$ such that $\mathcal{F}^u_f $ and $\mathcal{F}^s_f$ define in $\mathcal{U}_x$ a local product structure.
%
%By absolute continuity of $\mathcal{F}^u_f $ inside $\mathcal{U}_x,$ (VAMOS ACREDITAR NISTO, POR ENQUANTO SEM PROVA) for a.e. $q \in \mathcal{U}_x$ we have
%
%$$ m_{\mathcal{F}^u_f(q)} \sim Leb_{{\mathcal{F}^u_f(q)}}, $$
%
%where $Leb_{{\mathcal{F}^u_f(q)}}$ is the measure induced by riemannian volume form of $\mathcal{F}^u_f(q).$
%
%Without lost generality, by absolute continuity  we can consider
%
%$$ m_{\mathcal{F}^u_f(x)} \sim Leb_{{\mathcal{F}^u_f(x)}}, $$
%
%and $Leb_{{\mathcal{F}^u_f(x)}}(B \cap \mathcal{U}_x)$ is full.
%
%
%
%
%Define $F^u(x) = \mathcal{F}^u_f(x) \cap \mathcal{U}_x \cap B, $
%
%
% For a point $z, y \in F^u(x) ,$ consider its pre-historis $\overline{a}(z),\overline{a}(y),$  using the uniform continuity of $\varphi$ we have $\displaystyle\lim_{n \rightarrow + \infty}d(\varphi(y_{-n}), \varphi(z_{-n})) \rightarrow 0,$ and by Cesaro convergence, we have:
%
%
%$$ \alpha := \displaystyle\lim_{n \rightarrow + \infty}\frac{1}{n}\sum_{i= 0}^n \varphi(y_{-i}) =  \displaystyle\lim_{n \rightarrow + \infty}\frac{1}{n}\sum_{i= 0}^n \varphi(z_{-i}). $$
%
%
%Note that, for every point $z \in p(C), $ we have the convergence of the average
%
%$$\frac{1}{n}\sum_{i= 0}^n \varphi(f^i(z))= \alpha.$$
%
%Now, consider the saturation of $F^u(x)$ by stable manifolds is $\mathcal{U}_x,$  Since $\mathcal{F}^s_f$ is transversally absolutely continuous we have
%
%$$m(sat^s(F^u(x)\cap \mathcal{U}_x) = m( \mathcal{U}_x) ,$$
%
%also for every $z' \in sat^s(F^u(x)\cap \mathcal{U}_x)$ we have
%
%$$ \overline{\varphi}(z') = \overline{\varphi}(z) = \alpha,$$
%
%where
%
%$$\overline{\varphi}(\cdot) = \lim_{n \rightarrow +\infty}\frac{1}{n} \sum_{i = 0}^{\infty} \varphi(f^i(\cdot)) $$
%
%where $z' \in \mathcal{F}^s_f(z), z \in F^u(x).$
%
%In particular $\overline{\varphi} $ is constant in $m$ a.e. $z \in \mathcal{U}_x.$ We have the following property, for every $x \in M,$ there is $\mathcal{U}_x$ such that $\overline{\varphi}|_{\mathcal{U}_x}$ is constant for $m$ a.e. $z \in \mathcal{U}_x.$
%
%
%
%
%
%To finish the proof, for each $c \in \mathbb{R}$ consider
%
%$$\mathcal{U}(c) = \displaystyle\bigcup_{\overline{\varphi}|\mathcal{U}_x \equiv c, \; m-a.e.} \mathcal{U}_x,$$
%
%Now, since  $M$ is connected and $M = \displaystyle\bigcup_{c \in \mathbb{R}} \mathcal{U}(c),$ we conclude that $\overline{\varphi}$ is constant in $M.$
%
%
%
%
%\begin{corollary}
%Let $f: M\rightarrow M$ be a $C^2-$  decomposable and  conservative  Strong Anosov Endomorphism, then $\overline{Per(f)} = \Omega(f) = M,$ in particular $f $ is transitive.
%\end{corollary}
%
%
%


%\begin{remark}[desnecess\'{a}rio] Se o lema anterior estiver certo, ainda que falhe esta coisa de folhea\c{c}\~{a}o inst\'{a}vel codificada, o caso linear do toro est\'{a} a salvo e muito provavelmente o caso Strong Anosov Endomorphisms tamb\'{e}m.
%\end{remark}

%\newpage
%
%
%
%\section{Ergodicity for $C^2-$Conservative Strongly Anosov Endomorphism}
%
%In this section we present the proof of the ergodicity for  $C^2-$Conservative Strong Anosov Endomorphism. For instance all of Anosov Endomorphism of a surface is a Strong Anosov Endomorphism, in particular Linear Anosov Endomorphism of the torus so is.
%
%
%
%
%Although $\mathbb{R}^n$ is not compact the properties above implies that $\widetilde{f} - D\widetilde{f}$ has uniform Lipschitz constant in $\mathbb{R}^n$ and $||D\widetilde{f}|| $ and $||D^2\widetilde{f}||$ are uniform bounded in $\mathbb{R}^n,$ so analogously to compact setting we have:
%

%
%
%

%
%
%
%
%
%

%
%
%
%\begin{proof}
%The proof consists in verify ergodicity by a following criterium:
%
%\begin{center}
%$(f,m)$ \textit{is ergodic} $\Leftrightarrow$ Given a $f$ \textit{invariant observable}  $\varphi \in\mathcal{ L}^1(\mathbb{T}^n) \rightarrow \mathbb{R},$ \textit{then} $\varphi$ \textit{is constant} $m-$ \textit{almost everywhere}.
%\end{center}
%
%
%
%It suffices to prove that, for any continuous observable $\varphi: M \rightarrow \mathbb{R},$ the projection
%
%$$\overline{\varphi}(x) = \lim_{n \rightarrow \infty}\frac{1}{n} \sum_{i=0}^{n-1} \varphi(f^i(x)),$$
%
%is constant in a.e. $x \in M.$
%
%
%Consider $\widetilde{m}$ the unique measure in $M^f, $ such that $p\widetilde{m} = m,$ such that $\widetilde{m}$ is $\theta^{f}$ invariant, like lemma above.
%
%
%By Ergodic Theorem applied to $(\theta^{f}, \widetilde{m})$ with observable $\varphi^{f},$ where $\varphi^f(\widetilde{x}) =  \varphi (x_0),$ we have that, there exists a $\widetilde{m}$ full set  $ C \subset M^f,$ such that, for every $\widetilde{x} \in C,$ holds:
%
%
%$$\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{i=0}^{n-1} \varphi^f((\theta^f)^i(\widetilde{x})) = \lim_{n \rightarrow \infty}\frac{1}{n} \sum_{i=0}^{n-1} \varphi^f((\theta^f)^{-i}(\widetilde{x})),  $$
%
%in other words, it means
%
%$$\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{i=0}^{n-1} \varphi(x_i) = \lim_{n \rightarrow \infty}\frac{1}{n} \sum_{i=0}^{n-1} \varphi(x_{-i}),  $$
%
%for every $\widetilde{x} = (\ldots, x_{-1}, x_0, x_1, \ldots) \in C.$
%
%Note that $X =  p(C)$ is a  $m-$ full measure set in $M.$ In fact,
%
%$$ m(p(C)) = \widetilde{m}(p^{-1}(p(C))) \geq \widetilde{m}(C) = 1. $$
%
%
%So,
%\begin{equation}\label{birk}
%\lim_{n \rightarrow \infty} \frac{1}{n}\sum_{i=0}^{n-1} \varphi(x_i) = \lim_{n \rightarrow \infty}\frac{1}{n} \sum_{i=0}^{n-1} \varphi(x_{-i}),
%\end{equation}
%
%for a.e. $x \in M.$
%
%














%--------------------------------------------------------------------------------------
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%\bibitem{BaBo}
%A. Baraviera and C. Bonatti,
%\newblock Removing zero Lyapunov Exponents.
%\newblock {\em Ergodic Theory of Dynamical Systems}, 23 :1655--1670, 2003.
%
%%\bibitem{Pes.Bar}
%%L.Barreira, Ya. Pesin.
%%\newblock Nonuniform Hyperbolicity Dynamics of Systems with Nonzero Lyapunov Exponents.
%%\newblock {Encyclopedia of Mathematics and its Applications}, Cambridge University Press, 2007.
%
%
%
%

%\bibitem{Br-Bu-Iv1}
%M. Brin, D. Burago, S. Ivanov,
%\newblock On partially hyperbolic of $3-$manifolds with commutative fundamental group .
%\newblock {\em Modern Dynamical Systems and Applications,} Boris Hasselblatt, M. Brin and Y. Pesin, eds,
%Cambridge Univ. Press, New York, 307--312, 2004.
%
%\bibitem{Brpe}
%M. Brin, Y. Pesin,
%\newblock Partially hyperbolic Dynamical systems.  , 1:177--212, 1974.
%\newblock {\em Izv. Acad. Nauk. SSSR,} vol.3, No. 1, 1--11, 2009.
%
%\bibitem{Br-Bu-Iv2}
%M. Brin, D. Burago, S. Ivanov,
%\newblock Dynamical  Coherence of Partially Hyperbolic Diffeomorfism of the $3-$Torus.
%\newblock {\em Journal of Modern Dynamical Systems and Applications,} vol.3, No. 1, 1--11, 2009.
%
%
%%\bibitem{PC}
%%P. Carrasco,
%%\newblock Compact Dynamical Foliations.
%%\newblock {\em Ph.D Thesis,} University of Toronto, 2010.
%
%
%
%
%
%
%
%\bibitem{GO}
%A. Gogolev,
%\newblock  How typical are pathological foliations in partially hyperbolic dynamics:
%an example.
%{\em Israel J. Math,} 187(2012),  493-507.
%
%%\bibitem{GoGu}
%%A. Gogolev, M.Guysinsky ,
%%\newblock $C^1-$differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus.
%%\newblock {\em Discrete contin. Dyn. Syst.,} (22), No.1-2, 183--200, 2008.
%
%
%%\bibitem{Gro}
%%M.Gromov.
%%\newblock Entropy, Homology and Semi-algebraic gemometry.
%%\newblock {Seminaire Bourbaki}, 663, 1985-1986.
%
%
%

%
%\bibitem{HP}
%M. Hirayama, Y. Pesin,
%\newblock Non-Absolutely Continuous Foliations.
%\newblock {\em Israel J. Math.,}160(2007), 173-187.
%

%
%\bibitem{HU}
%A. Hammerlindl, R. Ures,
%\newblock Ergodicity and Partial Hyperbolicity on the Three Torus.
%\newblock {\em Coomunication in Contemporary Mathematics.,}  2013,
%
%
%\bibitem{SW00}
%M. Shub, A. Wilkinson.
%\newblock Pathological foliations and removable zero exponents.
%\newblock {\em Inv. Math.} 139, no. 3, 495-508, 2000.
%
%
%
%
%
%
%\bibitem{L}
%F. Ledrappier.
%\newblock Proprietes ergodiques des mesures de Sina\"{\i}.
%\newblock{\em Publ. Math. I.H.E.S.}, 59:163�188,
%1984.
%
%%\bibitem{LY}
%%F. Ledrappier, L.-S. Yang.
%%\newblock The metric entropy of diffeomorphisms. I. Characterization
%%of measures satisfying Pesin�s entropy formula.
%%\newblock{\em Ann. of Math.}, (2) 122, No. 3, 509-539, 1985.
%
%
%
%%\bibitem{DeL}
%%R. de la Llave,
%%\newblock Smooth conjugacy and {S}{R}{B} measures for uniformly and nonuniformly
%%hyperbolic systems,
%%\newblock{\em Communications in Mathematical Physics} 150,
%%289--320, 1992.
%%
%%\bibitem{Ma}
%%R.Ma��.
%%\newblock Teoria Erg�dica.
%%\newblock {Instituto de Matem�tica Pura e Aplicada},  1983.
%
%
%
%\bibitem{MT}
%F. Micena, A. Tahzibi.
%\newblock Regularity of foliations and Lyapunov exponents for partially hyperbolic Dynamics.
%\newblock {\em Nonlinearity,}  (2013), no. 33, 1071-1082.
%
%
%
%
%\bibitem{Mic}
%F. Micena.
%\newblock Avan\c{c}os em din\^{a}mica parcialmente hiperb\'{o}lica e entropia para sistema iterado de fun\c{c}\~{o}es.
%\newblock {tese de doutorado}, Universidade de S\~{a}o Paulo  2011.
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%
%
%
%
%
%\bibitem{PS}
%Ya. Pesin, Ya.Sinai.
%\newblock  Gibbs measures for partially hyperbolic attractors.
%\newblock {\em Ergodic Theory
%Dynam. Systems,} 2 (1983), no. 3-4, 417-438.
%
%
%
% \bibitem{PT}
% G. Ponce, A. Tahzibi,
% \newblock Central Lyapunov exponents of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$,
% \newblock {\em To appear in Proceedings of American Math. Society, 2013}.
%
%
%\bibitem{PTV}
% G. Ponce, A. Tahzibi, R. Var$\widetilde{a}$o
% \newblock Minimal yet measurable foliations,
% \newblock {\em preprint, 2013}.
%
%
%\bibitem{PVW}
%C. Pugh, M. Viana, and A. Wilkinson.
%\newblock Absolute continuity of foliations. In preparation.
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%
%
%\bibitem{RHRHU2}
%F. Rodriguez Hertz, M.  Rodriguez Hertz and R. Ures,
%\newblock A survey of partially hyperbolic dynamics. Partially hyperbolic dynamics, laminations, and Teich\"uller flow
%\newblock {\em Fields Inst. Commun.,}  51,  35--87, 2007.
%
%
%\bibitem{SW00}
%M. Shub, A. Wilkinson.
%\newblock Pathological foliations and removable zero exponents.
%\newblock {\em Inv. Math.} 139, no. 3, 495-508, 2000.
%
%
%\bibitem{SX08}
%R. Saghin, Zh. Xia.
%\newblock Geometric expansion, Lyapunov exponents and foliations.
%\newblock {\em Ann.
%Inst. H. Poincar\'{e}} 26, no. 2, 689-704, 2009.
%
%
%\bibitem{Var}
%R. Var$\widetilde{a}$o,
%\newblock Center foliation: absolute continuity, disintegration and rigidity.
%\newblock {\em preprint}, 2013.
%
%
%
%
%
%
%
%
%
%
%
%
%



\end{thebibliography}

\end{document}

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{AVW %A. Avila, M. Viana and A. Wilkinson, %\newblock Absolute continuity, Lyapunov exponents, %and rigidity I: geodesic flows. Arxive: 1110.2365. % % %
1412.0629.extracted_bib
{BaBo %A. Baraviera and C. Bonatti, %\newblock Removing zero Lyapunov Exponents. %\newblock {\em Ergodic Theory of Dynamical Systems, 23 :1655--1670, 2003. % %%
1412.0629.extracted_bib
{Pes.Bar %%L.Barreira, Ya. Pesin. %%\newblock Nonuniform Hyperbolicity Dynamics of Systems with Nonzero Lyapunov Exponents. %%\newblock {Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2007. % % % % %
1412.0629.extracted_bib
{Br-Bu-Iv1 %M. Brin, D. Burago, S. Ivanov, %\newblock On partially hyperbolic of $3-$manifolds with commutative fundamental group . %\newblock {\em Modern Dynamical Systems and Applications, Boris Hasselblatt, M. Brin and Y. Pesin, eds, %Cambridge Univ. Press, New York, 307--312, 2004. % %
1412.0629.extracted_bib
{Brpe %M. Brin, Y. Pesin, %\newblock Partially hyperbolic Dynamical systems. , 1:177--212, 1974. %\newblock {\em Izv. Acad. Nauk. SSSR, vol.3, No. 1, 1--11, 2009. % %
1412.0629.extracted_bib
{Br-Bu-Iv2 %M. Brin, D. Burago, S. Ivanov, %\newblock Dynamical Coherence of Partially Hyperbolic Diffeomorfism of the $3-$Torus. %\newblock {\em Journal of Modern Dynamical Systems and Applications, vol.3, No. 1, 1--11, 2009. % % %%
1412.0629.extracted_bib
{PC %%P. Carrasco, %%\newblock Compact Dynamical Foliations. %%\newblock {\em Ph.D Thesis, University of Toronto, 2010. % % % % % % % %
1412.0629.extracted_bib
{GO %A. Gogolev, %\newblock How typical are pathological foliations in partially hyperbolic dynamics: %an example. %{\em Israel J. Math, 187(2012), 493-507. % %%
1412.0629.extracted_bib
{GoGu %%A. Gogolev, M.Guysinsky , %%\newblock $C^1-$differentiable conjugacy of Anosov diffeomorphisms on three dimensional torus. %%\newblock {\em Discrete contin. Dyn. Syst., (22), No.1-2, 183--200, 2008. % % %%
1412.0629.extracted_bib
{Gro %%M.Gromov. %%\newblock Entropy, Homology and Semi-algebraic gemometry. %%\newblock {Seminaire Bourbaki, 663, 1985-1986. % % % % %
1412.0629.extracted_bib
{HP %M. Hirayama, Y. Pesin, %\newblock Non-Absolutely Continuous Foliations. %\newblock {\em Israel J. Math.,160(2007), 173-187. % % %
1412.0629.extracted_bib
{HU %A. Hammerlindl, R. Ures, %\newblock Ergodicity and Partial Hyperbolicity on the Three Torus. %\newblock {\em Coomunication in Contemporary Mathematics., 2013, % % %
1412.0629.extracted_bib
{SW00 %M. Shub, A. Wilkinson. %\newblock Pathological foliations and removable zero exponents. %\newblock {\em Inv. Math. 139, no. 3, 495-508, 2000. % % % % % % %
1412.0629.extracted_bib
{L %F. Ledrappier. %\newblock Proprietes ergodiques des mesures de Sina\"{\i. %\newblock{\em Publ. Math. I.H.E.S., 59:163�188, %1984. % %%
1412.0629.extracted_bib
{LY %%F. Ledrappier, L.-S. Yang. %%\newblock The metric entropy of diffeomorphisms. I. Characterization %%of measures satisfying Pesin�s entropy formula. %%\newblock{\em Ann. of Math., (2) 122, No. 3, 509-539, 1985. % % % %%
1412.0629.extracted_bib
{DeL %%R. de la Llave, %%\newblock Smooth conjugacy and {S{R{B measures for uniformly and nonuniformly %%hyperbolic systems, %%\newblock{\em Communications in Mathematical Physics 150, %%289--320, 1992. %% %%
1412.0629.extracted_bib
{Ma %%R.Ma��. %%\newblock Teoria Erg�dica. %%\newblock {Instituto de Matem�tica Pura e Aplicada, 1983. % % % %
1412.0629.extracted_bib
{MT %F. Micena, A. Tahzibi. %\newblock Regularity of foliations and Lyapunov exponents for partially hyperbolic Dynamics. %\newblock {\em Nonlinearity, (2013), no. 33, 1071-1082. % % % % %
1412.0629.extracted_bib
{Mic %F. Micena. %\newblock Avan\c{cos em din\^{amica parcialmente hiperb\'{olica e entropia para sistema iterado de fun\c{c\~{oes. %\newblock {tese de doutorado, Universidade de S\~{ao Paulo 2011. % % % % % % %
1412.0629.extracted_bib
{PS %Ya. Pesin, Ya.Sinai. %\newblock Gibbs measures for partially hyperbolic attractors. %\newblock {\em Ergodic Theory %Dynam. Systems, 2 (1983), no. 3-4, 417-438. % % % %
1412.0629.extracted_bib
{PT % G. Ponce, A. Tahzibi, % \newblock Central Lyapunov exponents of partially hyperbolic diffeomorphisms of $\mathbb{T^3$, % \newblock {\em To appear in Proceedings of American Math. Society, 2013. % % %
1412.0629.extracted_bib
{PTV % G. Ponce, A. Tahzibi, R. Var$\widetilde{a$o % \newblock Minimal yet measurable foliations, % \newblock {\em preprint, 2013. % % %
1412.0629.extracted_bib
{PVW %C. Pugh, M. Viana, and A. Wilkinson. %\newblock Absolute continuity of foliations. In preparation. % % % %
1412.0629.extracted_bib
{RHRHU2 %F. Rodriguez Hertz, M. Rodriguez Hertz and R. Ures, %\newblock A survey of partially hyperbolic dynamics. Partially hyperbolic dynamics, laminations, and Teich\"uller flow %\newblock {\em Fields Inst. Commun., 51, 35--87, 2007. % % %
1412.0629.extracted_bib
{SW00 %M. Shub, A. Wilkinson. %\newblock Pathological foliations and removable zero exponents. %\newblock {\em Inv. Math. 139, no. 3, 495-508, 2000. % % %
1412.0629.extracted_bib
{SX08 %R. Saghin, Zh. Xia. %\newblock Geometric expansion, Lyapunov exponents and foliations. %\newblock {\em Ann. %Inst. H. Poincar\'{e 26, no. 2, 689-704, 2009. % % %
1412.0629.extracted_bib
{Var %R. Var$\widetilde{a$o, %\newblock Center foliation: absolute continuity, disintegration and rigidity. %\newblock {\em preprint, 2013. % % % % % % % % % % % % %