Eigenvalues of minimal Cantor systems
Fabien Durand
Alexander Frank
Alejandro Maass
In this article we provide necessary and sufficient conditions that a complex number should verify to be a measurable or a continuous eigenvalue of a minimal Cantor system. These conditions are established from the combinatorial information carried by the Bratteli-Vershik representations of such systems. Moreover we explore how modifications of the local order of these representations spoil the group of eigenvalues and their quality of being continuous or not. We show that, giving precise bounds, if there are not too many of these modifications then the group of continuous eigenvalues does not change for the resulting systems and too many could change this group significantly.
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\begin{document}

\title
{Eigenvalues of minimal Cantor systems}

\author{Fabien Durand}
\address{Laboratoire Ami\'enois
de Math\'ematiques Fondamentales et Appliqu\'ees, CNRS-UMR 7352, Universit\'{e} de Picardie Jules Verne, 33 rue Saint Leu, 80000 Amiens,
France.} \email{fabien.durand@u-picardie.fr}

\author{Alexander Frank}
\address{Departamento de Ingenier\'{\i}a
Matem\'atica and Centro de Modelamiento Ma\-te\-m\'a\-ti\-co, CNRS-UMI 2807, Universidad de Chile, Beauchef 851, Santiago, Chile.}
\email{afrank@dim.uchile.cl}

\author{Alejandro Maass}
\address{Departamento de Ingenier\'{\i}a
Matem\'atica and Centro de Modelamiento Ma\-te\-m\'a\-ti\-co, CNRS-UMI 2807, Universidad de Chile, Beauchef 851, Santiago,
Chile.}\email{amaass@dim.uchile.cl}

\subjclass[2010]{Primary: 54H20; Secondary: 37B20} \keywords{Minimal Cantor systems, Bratteli-Vershik representations, eigenvalues}

\thanks{The first author was partially supported by the ANR programs FAN and DynA3S, and MathAmSud DYSTIL. The second and third authors were partially supported by grants Basal-CMM, Fondap 15090007 and Mecesup UCH 0607. We acknowledge invitations from U. Picardie Jules Verne and CNRS where part of this work was developed.}

\begin{abstract}
In this article we provide necessary and sufficient conditions that a complex number should verify to be a measurable or a continuous eigenvalue of a minimal Cantor system. 
These conditions are established from the combinatorial information carried by the Bratteli-Vershik representations of such systems. 
Moreover we explore how modifications of the local order of these representations spoil  the group of eigenvalues and their quality of being continuous or not.
We show that, giving precise bounds, if there are not too many of these modifications then the group of continuous eigenvalues does not change for the resulting systems and too many could change this group significantly.
\end{abstract}

\date{March 23, 2015}

\maketitle

\markboth{Fabien Durand, Alexander Frank, Alejandro Maass}{Eigenvalues of finite rank minimal Cantor systems}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The spectral theory of dynamical systems and in particular the study of eigenvalues of topological dynamical systems, either from a measure theoretical or a topological perspective, is a fundamental topic in ergodic theory, allowing to understand mixing properties and to characterise the Kronecker and maximal equicontinuous factors, among others. Particularly interesting and rich has been the study of eigenvalues and weakly mixing property of classical systems like interval exchange transformations \cite{nogueirarudolph, avilaforni, ferenczizamboniI, ferenczizamboniII} or other systems arising from translations on surfaces \cite{aviladelecroix}. From the symbolic dynamics point of view most of these systems have a representation as minimal Cantor systems of finite topological rank, {\it i.e.}, there is a symbolic extension that can be represented by a Bratteli-Vershik system such that the number of Kakutani-Rohlin towers per level is globally bounded. To characterise eigenvalues of the original systems it is enough to consider this class of Cantor systems.   
Of course, a general approach like this assumes that the particular nature and information carried by the original dynamics can be effectively translated into concrete properties of a ``good'' Kakutani-Rohlin representation, what is not evident. Nevertheless, good representations for interval exchange transformations and, in particular, irrational rotations of the torus have already been proposed (see \cite{gjerdejohansen,dartnelldurandmaass}). 


With these examples in mind, our main motivation is to provide general necessary and sufficient conditions for a complex number to be the eigenvalue, either continuous or measurable, of a minimal Cantor system of finite topological rank and more generally to drop the condition on the rank. In particular, we also want these conditions to be useful for proving weakly mixing properties, so the absence of eigenvalues. 


This problem for different subclasses of minimal Cantor systems of finite topological rank has been considered since the pioneer works of Dekking \cite{dekking}  and Host \cite{host}. There, it has been stated that eigenvalues of primitive substitution dynamical systems are always associated to continuous eigenfunctions, thus the maximal equicontinuous and measure-theoretical Kronecker factors coincide. Implicitly, both works give conditions to be an eigenvalue, however the complete characterisation of eigenvalues for substitution dynamical systems was given in \cite{ferenczimauduitnogueira}. Later, necessary and sufficient conditions to characterise continuous and measurable eigenvalues of linearly recurrent minimal Cantor systems are provided in \cite{lr} and \cite{necesariasuficiente}. These conditions are very effective and rely on the combinatorial data carried by the Bratteli-Vershik representations. Even if linearly recurrent systems are natural from the symbolic dynamics point of view (see \cite{du1,du2}), this class could be considered ``small'', meaning that in many classical examples, like interval exchange transformations, only a few maps have a symbolic representation of this kind. In fact, beyond other very particular features of the Cantor representations of concrete systems, most of them are of finite topological rank and not linearly recurrent. There are few general results concerning eigenvalues of minimal Cantor systems of finite topological rank. Some conditions either necessary or sufficient are given for being topological or measurable eigenvalues in \cite{rangofinito} and a detailed study of eigenvalues of Toeplitz systems of finite topological rank was recently given in \cite{dfm}. This last work motivates the ideas of the current work.  


In this article we provide necessary and sufficient conditions for being a measurable eigenvalue of a minimal Cantor system of finite topological rank (Theorem \ref{theo:cns_sinrho} and \ref{theo:cnsvpmedible}). 
For continuous eigenvalues we succeeded to drop the finite rank hypothesis (Theorem \ref{theo:nec-suff-cont}). 
In its conception, the conditions are very similar to those proposed for linearly recurrent systems. They are given in the form of the convergence of some series or special sequences and only depend on the combinatorial data provided by the  Bratteli-Vershik representations. The main difference here is that we need to include in an algebraic way the information of the local orders carried by these representations. Even if general, the drawback of these conditions is that they depend on a non trivial computation. In order to illustrate its effectiveness and use we consider two families of examples: Toeplitz systems (already developed in \cite{dfm}) and some symbolic extensions of irrational rotations of the interval. 
This last class, even if simple, allows to see the amount of information needed to compute eigenvalues. 
Finally, we use our  necessary and sufficient condition to be a continuous eigenvalue to prove that by doing some controlled modifications of the local orders of a Bratteli-Vershik system one can spoil or not the group of eigenvalues and their quality of being continuous or not with the exception of the continuous eigenvalues that are roots of the unity.
In fact, we recover one of the main results in \cite{ormes} asserting that within a strong orbit equivalence class all systems have the same continuous eigenvalues that are roots of the unity.
As an application we produce, starting from any minimal Cantor system without continuous eigenvalues that are roots of the unity, a strong orbit equivalent system that is topologically weakly mixing and which shares the Kronecker factor with the original system.
In \cite{sadunpriebe} is developed such an example in the context of tiling systems. More results concerning eigenvalues of tiling systems can be found in \cite{solomyak,cortesgambaudomaass,alistecoronel} and the reference therein.

The article is organised as follows. In section \ref{sec:definitions} we provide the main definitions concerning eigenvalues of dynamical systems and Bratteli-Vershik representations. Section \ref{sec:continuous} is devoted to the main result in the continuous case (Theorem \ref{theo:nec-suff-cont}), together with some interesting applications, and Section \ref{sec:measurable} to the measurable case results (Theorem \ref{theo:cns_sinrho} and Theorem \ref{theo:cnsvpmedible}). Finally, in Section \ref{examples} we develop two examples illustrating our main results. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Definitions and notation}
\label{sec:definitions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Dynamical systems and eigenvalues}
A \emph{topological dynamical system}, or just dynamical system, is a compact Hausdorff space $X$ together with a homeomorphism 
$T:X\rightarrow X$. We use the notation $\left( X,T\right)$. If $X$ is a Cantor set 
({\it i.e.}, $X$ has a countable basis of closed and open sets and it has no isolated points) we say that the system is Cantor. A dynamical system is \emph{minimal} if all orbits are dense in $X$,
or equivalently the only non empty closed invariant set is $X$.

A complex number $\lambda$ is a {\it continuous eigenvalue} of $(X,T)$ if there exists a continuous function $f : X\to \CC$, $f\not = 0$, such
that $f\circ T = \lambda f$; $f$ is called a {\it continuous eigenfunction} (associated to $\lambda$). The system is \emph{topologically weakly mixing} if it has no non constant continuous eigenfunctions.
Let $\mu$ be a $T$-invariant probability measure, {\it i.e.}, $T\mu = \mu$, defined on the Borel $\sigma$-algebra of $X$. A complex number $\lambda$ is an {\it eigenvalue} of the
dynamical system $(X,T)$ with respect to $\mu$ if there exists $f\in L^2 (X,\mu)$, $f\not = 0$, such that $f\circ T = \lambda f$; $f$ is
called an {\it eigenfunction} (associated to $\lambda$). If $\mu$ is ergodic, then every eigenvalue has modulus 1 and every eigenfunction has a constant modulus $\mu$-almost surely. Of course, continuous eigenvalues are eigenvalues. The system is \emph{weakly mixing} for $\mu$ if it has no non constant eigenfunctions.

If $\lambda=\exp(2i\pi\alpha)$ is either a continuous or measurable eigenvalue with $\alpha$ an irrational number we say that $\lambda$ is an irrational eigenvalue. 

\subsection{Bratteli-Vershik representations}
Let $(X,T)$ be a minimal Cantor system. It can be represented by an ordered Bratteli diagram together with the Vershik transformation acting on it. For details on this theory see \cite{hps} or \cite{review}. This couple is called a Bratteli-Vershik representation of the system. We give a brief outline of this construction emphasizing the notations in this paper.

\subsubsection{Bratteli diagrams}
A Bratteli diagram is an infinite graph $\left( V,E\right)$ which consists of a vertex set $V$ 
and an edge set $E$, both of
which are divided into levels $V=V_{0}\cup V_{1}\cup \ldots$, $E=E_{1}\cup E_{2}\cup \ldots$ 
and all levels are pairwise disjoint. 
The set $V_{0}$ is a singleton $\{v_{0}\}$ and for all $n\geq 1$ edges in $E_{n}$ join vertices in $V_{n-1}$ to vertices in $V_{n}$. If $e\in E$ connects $u\in V_{n-1}$ with $v \in V_n$ we write $\source(e)=u$ and $\range(e)=v$, where $\source:E_n\to V_{n-1}$ and $\range:E_n\to V_{n}$ are the source and range maps, respectively. 
It is also required that $\source^{-1}(v)\not = \emptyset$ for all $v \in V$ and that 
$\range^{-1}(v)\not = \emptyset$ for all $v \in V\setminus V_0$.
For all $n\geq 1$
we set $\#V_{n}=d_{n}$ and we write $V_n=\{1,\ldots,d_n\}$ to simplify notations.   

Fix $n\geq 1$. We call \emph{level} $n$ of the diagram to the subgraph consisting of the vertices in $V_{n-1}\cup V_{n}$ 
and the edges $E_{n}$ between these vertices. Level $1$ is called the \emph{hat} of the Bratteli diagram. 
We describe the edge set $E_n$ using a $V_{n-1}\times V_{n}$ incidence matrix $M_{n}$ for which its $(u,v)$ entry is the number of edges in $E_{n}$ joining vertex $u \in V_{n-1}$ with vertex $v \in V_{n}$. We also set $P_{n}=M_{2}\cdots M_{n}$ with the convention that $P_{1}=I$, where $I$ denotes the identity matrix. The number of paths joining $v_{0} \in V_{0}$ and a vertex $v\in V_{n}$ is given by coordinate $v$ of the \emph{height row vector} $h_{n}=\left(h_n(u) ; u \in V_{n} \right) \in \NN ^{d_{n}}$. Notice that $h_{1}=M_{1}$ and $h_{n}=h_{1}P_{n}$. 

We also consider several levels at the same time. 
For integers $0 \leq m< n$ we denote by $E_{m,n}$ the set of all paths in the graph joining vertices of  
$V_{m}$ with vertices of $V_{n}$. We define matrices $P_{m,n}=M_{m+1} \cdots M_{n}$ with the convention that $P_{n,n}=I$
for $1\leq m \leq n$. Clearly, coordinate $\matrizp{m}{n}{u}{v}$ of matrix $P_{m,n}$ is the number of paths in $E_{m,n}$ from vertex $u \in V_{m}$ to vertex $v \in V_{n}$. 
It can be verified that $h_{n}=h_{m}P_{m,n}$. 

We need to notice that the incidence matrices defined above correspond to the transpose of the matrices defined at the classical reference in this theory \cite{hps}. This choice, that in our opinion is more mnemotechnical, is done to simplify the reading of the article.  

\subsubsection{Ordered Bratteli diagrams and Bratteli-Vershik representations}
An \emph{ordered} Bratteli diagram is a triple \( B=\left( V,E,\preceq \right) \), where \( \left( V,E\right)  \) is a Bratteli diagram and \( \preceq  \) is a partial ordering on \( E \) such that: edges
\( e \) and \( e' \) are comparable if and only if $\range(e)=\range(e')$.
This partial ordering naturally defines maximal and minimal edges and paths. Also, 
the partial ordering of $E$ induces another one on paths of $E_{m,n}$, where for $0 \leq m < n$:  
$\left(e_{m+1},\ldots,e_{n}\right) \preceq \left(f_{m+1},\ldots ,f_{n}\right)$ if and only if 
there is $m+1\leq i\leq n$ such that $e_{j}=f_{j}$ for $i<j\leq n$ and $e_{i}\preceq f_{i}$.

Given a strictly increasing sequence of integers 
$\left(n_{k}\right)_{k\geq 0}$ with $n_{0}=0$ one defines the \emph{contraction} or \emph{telescoping} of
$B=\left(V,E,\preceq \right)$ with respect to $\left(n_{k} \right)_{k\geq 0}$ as 
$$\left(\left(V_{n_{k}}\right)_{k\geq
0},\left( E_{n_{k},n_{k+1}}\right)_{k\geq 0},\preceq \right), $$ where $\preceq$ is the order induced in each set of edges 
$E_{n_{k},n_{k+1}}$. The converse operation is called {\it microscoping} (see \cite{hps} for more details).
\smallskip

Given an ordered Bratteli diagram \( B=\left( V,E,\preceq \right) \) one defines \( X_{B} \) as the set of infinite paths \( \left(
x_{1},x_{2},\ldots \right)  \) starting in \( v_{0} \) such that $\range(x_n)=\source(x_{n+1})$
for all \( n\geq 1 \). We topologize \( X_{B} \) by postulating a basis of open sets, namely the family of \emph{cylinder sets}
$$
\left[ e_{1},e_{2},\ldots ,e_{n}\right]
=\left\{ \left( x_{1},x_{2},\ldots \right) \in X_{B} \textrm{ } ; \textrm{ }
x_{i}=e_{i},\textrm{ for }1\leq i\leq n\textrm{ }
\right\} .$$

Each \( \left[ e_{1},e_{2},\ldots ,e_{n}\right]  \) is also closed, as is
easily seen, and so \( X_{B} \) is a compact, totally disconnected metrizable space.

When there is a unique point  \( \left( x_{1},x_{2},\ldots \right) \in X_{B} \) such that \( x_{n} \) is (locally) maximal for any \( n\geq 1 \) and a unique
point \( \left( y_{1},y_{2},\ldots \right) \in X_{B} \) such that \( y_{n} \) is (locally) minimal for any \(n \geq 1 \), one says that \( B=\left(
V,E,\preceq \right)  \) is a \emph{properly ordered} Bratteli diagram. Call these particular points \( x_{\mathrm{max}} \) and \(
x_{\mathrm{min}} \) respectively. In this case one defines the dynamic \( V_{B} \) over \( X_{B} \) called the \emph{Vershik map}. Let \( x=\left( x_{1},x_{2},\ldots \right) \in X_{B}\setminus \left\{ x_{\mathrm{max}}\right\}  \) and let \(
n\geq 1 \) be the smallest integer so that \( x_{n} \) is not a maximal edge. Let \( y_{n} \) be the successor of \( x_{n} \) for the local order and \( \left(
y_{1},\ldots ,y_{n-1}\right)  \) be the unique minimal path in \( E_{0,n-1} \) connecting \( v_{0} \) with the initial vertex of \( y_{n} \). One
sets \( V_{B}\left( x\right) =\left( y_{1},\ldots ,y_{n-1},y_{n},x_{n+1},\ldots \right)  \) and \( V_{B}\left( x_{\mathrm{max}}\right)
=x_{\mathrm{min}} \).

The dynamical system \( \left( X_{B},V_{B}\right)  \) is minimal. It is called the \emph{Bratteli-Vershik system} generated by \( B=\left( V,E,\preceq \right)
\). The dynamical system induced by any telescoping of \( B \) is topologically conjugate to \( \left( X_{B},V_{B}\right)  \). In \cite{hps} it
is proved that any minimal Cantor system \( \left( X,T\right)  \) is topologically conjugate to a Bratteli-Vershik system \( \left(
X_{B},V_{B}\right)  \). One says that \( \left( X_{B},V_{B}\right) \) is a \emph{Bratteli-Vershik representation} of \( \left( X,T\right)  \).
In what follows we identify $(X,T)$ with any of its Bratteli-Vershik representations.

\subsubsection{Minimal Cantor systems of finite topological rank}

A minimal Cantor system is of finite (topological)  rank if it admits a  Bratteli-Vershik representation such that the number of vertices per level is uniformly bounded by some integer $d$. The minimum possible value of $d$ is called the \emph{topological rank} of the system. We observe that topological and measure theoretical finite rank  notions are completely different. For instance, systems of topological rank one correspond to odometers, whereas in the measure theoretical sense there are rank one systems that are expansive as classical Chacon's example. 

To have a better understanding of the dynamics of a minimal Cantor system, and in particular 
to understand its group of eigenvalues, one needs to work with a ``good'' Bratteli-Vershik representation. 
So we consider representations such that:

\smallskip

(H1) The entries of $h_{1}$ are all equal to $1$.

(H2) For every $n\geq 2$, $M_{n}>0$ (this is called a \emph{simple} Bratteli-Vershik diagram).

(H3) For every $n\geq 2$, all maximal edges of $E_n$ start in the same vertex of $V_{n-1}$. 
\medskip

It is classical to prove that this reduction is possible, in particular (H3) can be deduced from Proposition 2.8 in \cite{hps}. A Bratteli-Vershik representation of a  minimal Cantor system $(X,T)$ verifying (H1), (H2) and (H3) will be called \emph{proper}. If the minimal Cantor system has finite rank $d$, in the definition of proper representation we will also consider:
\medskip

(H4) For every $n\geq 2$, $d_{n}$ is equal to $d$.
\medskip

In this last case, to simplify notations and avoid the excessive use of indexes, we will identify $V_{n}$ with $\{1,\ldots, d\}$ for all $n\geq 1$. The level $n$ will be clear from the context. It is not difficult to prove that a  minimal Cantor system of topological finite rank $d$ has a proper representation (see \cite{dfm} for an outline of the proof). 


A minimal Cantor system is \emph{linearly recurrent} if it admits a proper Bratteli-Vershik representation such that the set $\{M_{n};n\geq 2\}$ is
finite. Clearly, linearly recurrent minimal Cantor systems are of finite rank (see \cite{dhs}, \cite{du1}, \cite{du2} and \cite{lr} for more details and properties of this class of systems).


\subsubsection{Associated Kakutani-Rohlin partitions}
Let $B=\left( V,E,\preceq \right)$ be a properly ordered Bratteli diagram and $(X,T)$ the associated minimal Cantor system. This diagram defines for each $n\geq 0$ a clopen {\it Kakutani-Rohlin} partition of $X$: for $n=0$, 
$\P_{0}=\{B_{0}(v_{0})\}$, where $B_{0}(v_{0})=X$, and for $n\geq 1$    
$$
\P_{n}=\{T^{-j}B_{n}(v); v \in V_n, \ 0 \leq j < h_n(v) \} \ ,
$$
where $ B_n(v) = [e_1 , \dots ,e_n]$ and $(e_1 , \dots ,e_n)$ is the unique maximal path from $v_0$ to vertex $v \in V_{n}$. For each $v \in V_n$ the set $\{T^{-j}B_n(v); 0 \leq j < h_n(v) \}$ is called the {\it tower} $v$ of $\P_{n}$. 
It corresponds to the set of all paths from $v_0$ to $v\in V_n$ (there are exactly $h_n(v)$ of such paths). 
Denote by $\T_{n}$ the $\sigma$-algebra generated by the partition $\P_{n}$.
The map $\tau_n: X \to V_n$ is
given by $\tau_n(x)=v$ if $x$ belongs to tower $v$ of $\P_{n}$. The entrance time of 
$x$ to $B_n({\tau_n(x)})$ is given by $r_n(x)=\min\{ j\geq 0; T^jx \in B_n({\tau_n(x)}) \}$. 


For each $x=(x_1,x_{2},\ldots) \in X$ and $0 \leq m<n$ define the row vector 
$s_{m,n}(x)\in \NN^{d_{m}}$, called the {\it suffix vector between $m$ and $n$} of $x$, by
$$
s_{m,n}(x,u)=\# \{e \in E_{m,n}; (x_{m+1},\ldots,x_n) \prec e, \source(e)=u \}
$$
at each coordinate $u\in V_m$, where $\prec$ stands for $\preceq$ and $\neq$ simultaneously.

For each $0 \leq m<n$, $u\in V_m$ and $v\in V_n$, define the set
$$
S_{m,n}(u,v)=\set{s_{m,n}(x)\talque \tau_m(x)=u \y \tau_n(x)=v}.
$$ 

A direct verification shows that the cardinality of $S_{m,n}(u,v)$ is equal to $\matrizp{m}{n}{u}{v}$, the number of elements in $E_{m,n}$ joining $u$ and $v$.
If necessary, to simplify notations we write $s_n(x)$ and $S_n(u,v)$ instead of $s_{n,n+1}(x)$ and $S_{n,n+1}(u,v)$ respectively.
 
A classical computation gives for all $n\geq 1$ (see for example \cite{necesariasuficiente})
\begin{align}
\label{eq:formulareturn} r_n(x)= s_0(x)+\sum_{i=1}^{n-1} \la s_i(x),h_{1}P_{i} \ra=s_0(x)+\sum_{i=1}^{n-1} \la s_i(x),h_{i} \ra \ ,
\end{align}
where $\la \cdot, \cdot \ra$ is the euclidean inner product. Observe that under  hypothesis (H1), {\it i.e.}, $h_{1}=(1,\ldots,1)$, we have $s_0(x)=0$. Similarly, it can be obtained the following general relation between entrance times and suffix vectors of $x$:
\begin{equation}\label{eq:formulareturn_mn}
r_n(x) = r_m(x) + \la s_{m,n}(x),h_m\ra ,
\end{equation}
for $1\leq m<n$. From this it follows that for $0\leq \ell < m < n$
\begin{equation}\label{eq:suma_sufijos}
\la s_{\ell,n}(x),h_{\ell}\ra = \la s_{\ell,m}(x),h_{\ell}\ra + \la s_{m,n}(x),h_{m}\ra,
\end{equation}
and particularly
\begin{equation}\label{eq:formulareturn_mn_suma}
\la s_{m,n}(x),h_{m}\ra = \sum_{i=m}^{n-1} \la s_{i}(x),h_{i}\ra.
\end{equation}

%%%%%%%%%%%%%%%%%%
\subsubsection{Invariant measures}
%%%%%%%%%%%%%%%%%%
\label{subsubsec:invmeas}
Let $\mu$ be an invariant probability measure of the system $(X,T)$ associated to a properly ordered Bratteli diagram 
$B$, like in the previous subsection. 
It is determined by the values assigned to $B_n(v)$ for all $n\geq 0$ and $v \in V_n$. Define the column vector
$\mu_{n}=(\mu_n(v)\ ; \ v \in V_{n})$ with $\mu_n(v)=\mu(B_n(v))$. 
A simple computation allows to prove the following useful relation:
\begin{equation}\label{eq:measure}
\mu_{m}=P_{m,n}\mu_{n}
\end{equation}
for integers $0\leq m < n$. Also, $\mu(\tau_{n}=v)=h_{n}(v) \mu_{n}(v)$ for all $n\geq 1$ and $v\in V_{n}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Clean Bratteli-Vershik representations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $B$ be a proper ordered Bratteli diagram of finite rank $d$ and $(X,T)$ the corresponding minimal Cantor system. 
Recall that in this case we identify $V_{n}$ with $\{1,\ldots, d\}$ for all $n\geq 1$. 
Then, by Theorem 3.3 in \cite{bkms}, there exist a telescoping of the diagram (which keeps the diagram proper) 
and $\delta_0 >0$ such that:  
\begin{enumerate}
\item For any ergodic measure $\mu$ there exists $I_{\mu} \subseteq \{1,\ldots,d\}$ verifying: 
\begin{enumerate}
\item $\mu(\tau_{n}=v) \geq \delta_0$ for every $v \in I_{\mu}$ and $n \geq 1$, and   
\item $\lim_{n\to +\infty} \mu(\tau_{n}=v)=0$ for every $v \not \in I_{\mu}$.
\end{enumerate}
\item If $\mu$ and $\nu$ are different ergodic measures then $I_{\mu}\cap I_{\nu}=\emptyset$.  
\end{enumerate}

When an ordered Bratteli diagram verifies the previous properties we say it is \emph{clean}. 
We remark that this is a modified version of the notion of \emph{clean} Bratteli diagram given in 
\cite{rangofinito} that is inspired by the results of \cite{bkms}. This property will be very 
relevant for formulating our main result in the measurable case. In \cite{bkms}, systems such that $I_{\mu}=\{1,\ldots,d\}$ for some ergodic measure 
$\mu$ are called of \emph{exact finite rank}. Clearly, those systems are suniquely ergodic. 
\smallskip


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Continuous eigenvalues of minimal Cantor systems}
\label{sec:continuous}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we show a necessary and sufficient condition for a complex number to be a continuous eigenvalue of a minimal Cantor system. The condition is given in terms of the combinatorial objects associated to a proper Bratteli-Vershik representation of the system. The proof follows the lines and some ideas developed to prove a general necessary condition in \cite{rangofinito}. 
Then, we show the utility of this condition to understand the role of the order in the existence of continuous eigenvalues. 
We give a condition allowing to modify the local orders without changing the group of continuous eigenvalues and we end with some comments on the dimension groups and the group of continuous eigenvalues of minimal Cantor systems. More examples are given in Section \ref{examples}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The necessary and sufficient condition}
%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $(X,T)$ be a minimal Cantor system given
by a proper Bratteli-Vershik representation and consider
$\lambda=\exp(2i\pi\alpha)$. The most general necessary and sufficient condition for $\lambda$ to be a continuous eigenvalue of $(X,T)$ states that 
$\lambda^{r_n(x)}$ converges uniformly (see Proposition 12 in \cite{necesariasuficiente}). In order to achieve the uniform convergence given by this condition several other simpler necessary conditions, relying on the combinatorics of the Bratteli-Vershik representations, have been proposed. 
We recall the necessary condition proved in {\cite{rangofinito}} that serve as motivation to the main result of this section.

\begin{theo}
\label{th:necessary-continuous} 
Let $(X,T)$ be a minimal Cantor system given
by a proper Bratteli-Vershik representation $B=\left( V,E,\preceq \right)$. 
If $\lambda=\exp(2i\pi \alpha)$ is a continuous
eigenvalue of $(X,T)$ then
$$
\displaystyle \sum_{n\ge 1} \vvert \alpha h_1 P_n \vvert = \sum_{n\ge 1} \vvert \alpha h_n \vvert <+\infty \ , 
$$
where $ \vvert\cdot  \vvert$ is the distance to the nearest integer.
\end{theo}

Assume $\lambda=\exp(2i\pi\alpha)$ is a continuous eigenvalue of $(X,T)$ 
as in Theorem \ref{th:necessary-continuous}. Then, for all $n\geq 1$ there exist a real vector $\eta_n$ and an integer vector $\nu_n$ such that
\begin{equation}
\label{eq:eta}
\alpha h_1 P_n = \alpha h_n =\eta_n + \nu_n \hbox{ and } \eta_n \xrightarrow[n\to +\infty]{}0 \ .
\end{equation}
Moreover, from the proof of Theorem 5 in \cite{rangofinito} we have that for all large enough $n$ 
\begin{equation}
\label{eq:eta1}
\eta_{n+1}=\eta_n M_{n+1} \text{ and } \nu_{n+1}=\nu_n M_{n+1} \ .  
\end{equation}
A classical computation allows to deduce the possible values for $\alpha$ from these two conditions. Fix large integers $1 \leq m <  n$ and multiply (the row vector) $\alpha h_m$ by (the column vector) $\mu_m$. From \eqref{eq:eta} we get,
\begin{align*}
\alpha &=\alpha h_m \cdot \mu_m = \eta_m \cdot \mu_m  +  \nu_m \cdot \mu_m
= \eta_m \cdot P_{m,n} \cdot \mu_{n} + \nu_m \cdot \mu_m = \eta_n \cdot \mu_{n} + \nu_m \cdot \mu_m, 
\end{align*}
where in the last equality we have used \eqref{eq:measure} and \eqref{eq:eta1}. Taking $n\to +\infty$ and using \eqref{eq:eta} we get that 
\begin{equation}
\label{eq:eta2}
\alpha= \nu_m \cdot \mu_m \text{ and } \eta_m \cdot \mu_m=0 \text{ for some (but in fact every) large enough $m\in \NN$}. 
\end{equation}
Below we will frequently call \eqref{eq:eta} and \eqref{eq:eta1} the necessary conditions to be continuous eigenvalue for $\lambda$. The main point here is that many possible values of $\lambda$ deduced from these conditions will not be continuous eigenvalues. This strongly rely on the local orders of the Bratteli-Vershik systems. 

The general necessary and sufficient condition we present below refines the one in Theorem \ref{th:necessary-continuous} but it incorporates the local orders of the Bratteli-Vershik representation. This is achieved by considering the suffix vectors $s_n(x)$ defined for each $x \in X$ at each level of the diagram. While submitting this article we remarked the similarity of this result with Theorem 4.1 in \cite{sadunpriebe}, where the authors provide a necessary and sufficient condition to be a continuous eigenvalue for a special class of fusion tilings. 

\begin{theo}
\label{theo:nec-suff-cont} Let $(X,T)$ be a minimal Cantor system given
by a proper Bratteli-Vershik representation $B=\left( V,E,\preceq \right)$. 
The following conditions are equivalent,
\begin{enumerate}
\item
$\lambda=\exp(2i\pi \alpha)$ is a continuous eigenvalue of $(X,T)$.
\item 
$\displaystyle \sum_{n\ge 1}  \max_{x\in X} \vvert\langle  s_n(x) ,\alpha h_n  \rangle\vvert  <+\infty
 $.
\item 
$\displaystyle \sum_{n\ge 1}  \max_{s \in S_n(u_n,u_{n+1})} \vvert\langle  s ,\alpha h_n  \rangle\vvert  <+\infty
 $ for any sequence of vertices $(u_{n}; n\geq 1)$ with $u_n \in V_n$.
\end{enumerate}
\end{theo}
\medskip

Clearly (2) and (3) are equivalent. We will need two preliminary lemmas to prove that (1) is equivalent with (2) and (3). The first one is almost identical to Lemma 4 in \cite{rangofinito} so we omit its simple proof. 
We assume $(X,T)$ is a minimal Cantor system as given in Theorem \ref{theo:nec-suff-cont}. 
\begin{lemma}
\label{lemma:twopoints} 
Assume the coefficients of the incidence matrices $(M_n;n\geq 2)$
are strictly positive. 
Let $(j_n; n \geq 1)$ be a sequence of positive integers such that  $j_{n+1} -
j_{n} \geq 3$ and $(e_{j_n+1}; n\geq 1)$ be a sequence of edges of the Bratteli diagram with $e_{j_n+1} \in E_{j_n+1}$.
Then, there exist points $x=(x_1,x_2,\ldots)$ and $y=(y_1,y_2,\ldots)$ in $X$ such that for all $\ n\geq 1$,
\begin{enumerate}
\item
$x_{j_n+1}=e_{j_n+1}$, $\range(x_{j_n+1})=\range(y_{j_n+1})$ and $s_{j_n}(y) =(0,\ldots,0)$ (or $y_{j_n+1}$ is a maximal edge);
\item
$x_j = y_j$ for $j_n+2\leq j \leq j_{n+1}-1$; 
\item
$s_{j_{n+1}-1} (x) = s_{j_{n+1}-1} (y) = (0, \ldots , 0)$ (or $x_{j_n+1}$ and $y_{j_n+1}$ are maximal edges).
\end{enumerate}
\end{lemma}
\medskip


\begin{lemma}
\label{prop:tendtozero} 
$\displaystyle
\max_{x\in X} |\langle  s_n(x) , \eta_n \rangle | \to_{n\to +\infty} 0. 
$
\end{lemma}
\begin{proof}
Since $\lambda$ is a continuous eigenvalue, as recalled above, from Proposition 12 in \cite{necesariasuficiente}
we have that the sequence $(\vvert \alpha r_n \vvert;n\in \NN)$ is a uniform Cauchy sequence, and, by definition, we have that $\alpha(r_{n+1}-r_n)= \langle s_n , \alpha h_n \rangle$ is a continuous function.

Fix $0< \varepsilon < 1/8$. By \eqref{eq:eta} and previous observation, there exists $n_0$ such that for all $n\geq n_0$ and all 
$x\in X$,  
$$
|| \eta_n || < \varepsilon  < 1/8 \hbox{ and } \vvert \langle s_n (x) , \eta_n \rangle \vvert< \varepsilon .
$$

Write $\langle s_n (x) , \eta_n \rangle = \varepsilon_n (x) + E_n(x)$ with $|\varepsilon_n (x)| < \varepsilon$ and $E_n(x)$ an integer vector. 
Notice that $(\varepsilon_n)_{n\in \NN}$ converges uniformly to $0$.

For $n\geq n_0$ consider the set $A_n = \{ x \in X  ; E_n(x) = 0 \}$.
Observe that this set is not empty (consider points with maximal edges at level $n+1$) and closed. Let us check that it is $T$-invariant. 
Consider $x \in A_n$. We have three cases: $s_n (x) = s_n (Tx)$, $s_n (x) = 0$ with  $s_n (x) \not = s_n (Tx)$, and 
$s_n (x) = s_n (Tx) + e$ for some vector $e$ from the canonical base. In the first case it is obvious that $Tx  \in A_n$ when $x \in A_n$. 

In the second one $(x_1,\ldots,x_{n+1})$ is formed by maximal edges and thus $(Tx)_{n+1}$ is a minimal edge. Therefore, 
$s_n(Tx)$ is the $v$-th column of $M_{n+1}$ minus the $u$-th canonical vector, where $u=\source((Tx)_{n+1})$ and $v=\range((Tx)_{n+1})$. Then, 
$$|\langle s_n(Tx), \eta_n \rangle |= | \sum_{u'\in V_n} \eta_n(u') M_{n+1}(u',v) - \eta_n(u)| = |  \eta_{n+1}(v) - \eta_n(u)| \leq \frac{1}{4} \ ,$$
where in the second equality we have used the relation in \eqref{eq:eta1}. Hence $E_n(Tx) = E_n(x) = 0$ and $Tx \in A_n$.

In the last case, 
\begin{align*}
|E_n(Tx ) - E_n(x)| = & |\varepsilon (x) -\varepsilon (Tx) + \langle s_n (T x) ,  \eta_n \rangle - \langle s_n (x) ,  \eta_n \rangle |\\
 \leq  & \frac 14 + |\langle e ,  \eta_n \rangle | \leq \frac 14 +  ||\eta_n|| \leq \frac 12 .
\end{align*}
Therefore $E_n(Tx) = E_n(x) = 0$ and $Tx \in A_n$.

By minimality, we obtain that $A_n=X$. This implies that for all $n \geq n_0$ 
$$
\vvert \langle  s_n(x) , \alpha h_n \rangle \vvert  = | \langle  s_n(x) ,\eta_n \rangle | = | \varepsilon_n (x) | \ ,
$$
which achieves the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{theo:nec-suff-cont}]
Let $\lambda=\exp(2i\pi \alpha)$ be a continuous eigenvalue of
$(X,T)$. From \eqref{eq:eta}, there exist a real vector 
$\eta_n$ and an integer vector $\nu_n$ such that
$$
\displaystyle
\alpha h_n = \eta_n + \nu_n \hbox{ and } \eta_n \xrightarrow[n\to +\infty]{}
0,
$$ 
for all $n\geq 1$.

Thus, to get condition (2) of the theorem it is enough to prove that the series $\sum_{n\geq 1} \max_{x\in X} | \langle s_n(x) , \eta_n \rangle |$ converges.

For $n\geq 1$ let $z^{(n)}=(z^{(n)}_1,z^{(n)}_2,\ldots) \in X$ be such that 
$$
| \langle  s_n(z^{(n)}) , \eta_n  \rangle |= \max_{x\in X}| \langle  s_n(x) , \eta_n  \rangle |  \ . 
$$
We set $e_{n+1} = z^{(n)}_{n+1}$ and ${\mathfrak s}_n = s_n(z^{(n)})$. So, it suffices to prove the following convergence,
\begin{align}
\label{series:suffisant}
\sum_{n\geq 1} | \langle {\mathfrak s}_n , \eta_n \rangle | < +\infty .
\end{align}
Let
$
I^+=\{n \geq 1; \langle {\mathfrak s}_{n}, \eta_n\rangle   \geq 0 \}, \ I^-=\{n \geq 1 ;
\langle {\mathfrak s}_{n}, \eta_n\rangle  <0\}.
$
To prove \eqref{series:suffisant}  we only need
to show that
$$
\sum_{n\in I^+} \langle {\mathfrak s}_{n},\eta_n\rangle   < +\infty \; \; \textrm{and}\; \;  -
\sum_{n\in I^-} \langle {\mathfrak s}_{n},\eta_n\rangle   < +\infty.
$$
Since the arguments in both cases are similar we only
prove the first one. Moreover, to prove $\sum_{n\in I^+}
\langle {\mathfrak s}_{n}, \eta_n\rangle  < +\infty$ we only show 
$
\sum_{n\in I^+\cap 3\NN}
\langle {\mathfrak s}_n,\eta_n\rangle   < +\infty.
$ 
In a similar way one proves the convergence of series 
$\sum_{n\in I^+\cap (3\NN+1)} \langle {\mathfrak s}_n,\eta_n\rangle$ and $\sum_{n\in I^+\cap (3\NN+2)} \langle {\mathfrak s}_n,\eta_n\rangle$.

Assume $I^+ \cap 3\NN$ is infinite, if not the result follows
directly. Order its elements: $1 <j_1 < j_2< \ldots <j_n<\ldots$.
From Lemma \ref{lemma:twopoints} there exist two points $x= (x_1,x_2, \ldots)$ and $y =(y_1,y_2,\ldots)$ in $X$ such that for all  $n\geq 1$, 

\begin{enumerate}
\item
$x_{j_n+1}=e_{j_n+1}$, $s_{j_n}(y) = (0,\ldots,0)$ and $\range(x_{j_n+1})=\range(y_{j_n+1})$;
\item $x_j = y_j$ for $j_n+2\leq j \leq j_{n+1}-1$; 
\item
$s_{j_{n+1}-1} (x) = s_{j_{n+1}-1} (y) = (0, \ldots , 0)$.
\end{enumerate}
We can also set $x_1=y_1,\ldots,x_{j_1-1}=y_{j_1-1}$ and $s(y_{j_1})=(0,\ldots,0)$. 
Hence we have, 
\begin{enumerate}
\item
$s_{j_n}(x) =  {\mathfrak s}_{j_n}$ and $ s_{j_n}(y)  = (0,\ldots,0)$;
\item
$s_{j-1}(x) - s_{j-1}(y) =  (0,\ldots,0)$ for $j_n+2\leq j \leq j_{n+1}-1$. 
\end{enumerate}

Now, from the definition of the return function in \eqref{eq:formulareturn} and properties of points $x$ and $y$ just constructed we get for all $m>1$, 

\begin{align*}
\alpha (r_m (x) - r_m (y)) 
&= \alpha 
\sum_{n\in \{1,\ldots,m-1\}} \langle s_n(x)-s_n(y), h_n\rangle  
\\
&= \alpha 
\sum_{n\in \{1,\ldots,m-1\}\cap I^+\cap 3\NN} \langle s_n(x)-s_n(y), h_n\rangle  
\\
&=\alpha \sum_{n\in \{1,\ldots,m-1\}\cap I^+\cap 3\NN} \langle {\mathfrak s}_{n}, h_n \rangle 
\\
&=\sum_{n\in \{1,\ldots,m-1\}\cap I^+\cap 3\NN} \langle {\mathfrak s}_{n},\eta_{n}+\nu_{n}\rangle \ .
\end{align*}

From Proposition 12 in \cite{necesariasuficiente} we have that $\alpha (r_m (x) - r_m (y))$ converges~$\mod \ZZ$ when $m \to +\infty$. Then $\sum_{n\in \{1,\ldots,m-1\}\cap I^+\cap 3\NN} \langle {\mathfrak s}_{n}, \eta_n\rangle $ converges~$\mod \ZZ$ when $m\to +\infty$ too. 
Finally, from Lemma \ref{prop:tendtozero}, $\langle {\mathfrak s}_{n},\eta_n\rangle $ tends to 0, hence the series $\sum_{n\in I^+\cap 3\NN} \langle {\mathfrak s}_{n},\eta_n\rangle $
converges. This proves that (1) implies (2). 
\medskip

Now we prove (2) implies (1). Suppose that 
$\sum_{n\ge 1}  \max_{x\in X} \vvert\langle  s_n(x) , \alpha h_n  \rangle  \vvert$ converges and let us prove that
$\lambda=\exp(2i\pi \alpha)$ is a continuous eigenvalue of $(X,T)$.

By Proposition 12 in \cite{necesariasuficiente}, it suffices to show that 
$(\lambda^{r_n};n\in \NN)$ converges uniformly, or, equivalently, that $(\alpha r_n;n\in \NN)$ is a Cauchy sequence~$\mod \ \mathbb{Z}$.
For $1 \leq m< n$ we have,
\begin{align*}
\left| \vvert \alpha r_n \vvert - \vvert \alpha r_m \vvert  \right| \leq & \vvert \alpha (r_n - r_m ) \vvert \\
 = & \vvert \alpha \left( \sum_{k=m+1}^n \langle s_k , h_k \rangle  \right) \vvert \\
 \leq &  \displaystyle \sum_{k=m+1}^n  \max_{x\in X} \vvert \langle  s_k(x) ,  \alpha h_k \rangle \vvert \ .
\end{align*}
Then, from condition (2) of the theorem one deduces that $\vvert \alpha r_n \vvert$ is a Cauchy sequence for the uniform convergence, which achieves the proof.
\end{proof}

\begin{rmrk}\label{rmrk:cond_suf_sin_simple}
In proving that condition (2) implies (1) it has not been used the fact that the diagram was proper. So, any Bratteli-Vershik representation of a minimal system $(X,T)$ which satisfies condition (2) of Theorem \ref{theo:nec-suff-cont} has $\lambda=\exp(2i\pi \alpha)$ as one of its continuous eigenvalues.
\end{rmrk}

As stated before, conditions \eqref{eq:eta} and \eqref{eq:eta1} allow to compute all possible values of $\alpha$ such that $\lambda=\exp(2i\pi \alpha)$ is a candidate to be a continuous eigenvalue. The main problem is to know whether they really correspond to continuous eigenvalues. This is related to the local orders of the Bratteli-Vershik representations and this is the point where Theorem \ref{theo:nec-suff-cont} plays a role. The following corollary (that is in the folklore) shows this fact: candidates that are roots of the unity are always continuous eigenvalues. Below we will comment about this result and strong orbit equivalence.

\begin{coro}\label{coro:rootsofunity}
Let $(X,T)$ be a minimal Cantor system given by a proper Bratteli-Vershik representation $B=(V,E,\preceq)$. Let $\alpha=p/q$ be a rational number verifying 
condition \eqref{eq:eta}. Then $\lambda=\exp(2i\pi p/q)$ is a continuous eigenvalue of $(X,T)$. In particular, it is enough that $q$ divides the height of the towers of $B$ at some level. 
\end{coro}
\begin{proof}
If $\alpha=p/q$ verifies \eqref{eq:eta} then necessarily $q$ divides the coordinates of the height vector $h_n$ and $\eta_n=0$ for all large enough $n \in \NN$. Hence, condition (2) in Theorem \ref{theo:nec-suff-cont} is verified and 
$\lambda=\exp(2i\pi p/q)$ is a continuous eigenvalue of $(X,T)$. 
\end{proof}

The case of non rational eigenvalues is more involving. In the next subsection we show some applications that illustrates this fact. 

We also stress the fact that computations involved in conditions of Theorem \ref{theo:nec-suff-cont} can be complicated and might require a lot of information of the Bratteli-Vershik representation of the system. 
If the minimal Cantor system is linearly recurrent the unique information required are the incidence matrices and heights since the suffix vectors are uniformly bounded as was already proved in \cite{necesariasuficiente}. Nevertheless, in a lot of relevant examples the information about the order, or equivalently the suffix vectors at each level, cannot be neglected.  

In order to show the kind of computations needed to use conditions of  Theorem \ref{theo:nec-suff-cont} we develop in Section \ref{examples} two simple cases. The first corresponds to Sturmian systems extending irrational rotations of the torus and the second one is the case of Toeplitz systems. In both cases continuous eigenvalues are well known. Our objective is not to produce new results but to show the kind of combinatorial information needed to compute such eigenvalues from Theorem \ref{theo:nec-suff-cont}. 

%%%%%%%%%%%%%%%%
\subsection{Modifying the local orders and continuous eigenvalues}
\label{sec:modifying}
%%%%%%%%%%%%%%%%
As was just mentioned the local orders do not play any role in the existence of continuous and measurable eigenvalues in the linearly recurrent case and this is also the case of continuous eigenvalues that are roots of the unity of any minimal Cantor system. The case of irrational continuous eigenvalues is much complicated and different orders on the same Bratteli diagram can produce completely different situations. Some examples illustrating this fact appear in \cite{rangofinito} and \cite{dfm}. Here we use Theorem \ref{theo:nec-suff-cont} to show how local orders modifications can leave invariant or significantly modify the group of continuous eigenvalues.

A first result shows how much the local orders can be modified without affecting the continuous eigenvalues. Before, let us define a $(\omega_n;n\geq 2)$-order modification of a Bratteli-Vershik representation $B=\left( V,E,\preceq \right)$, where $(\omega_n;n\geq 2)$ is a sequence of positive integers.
It is a Bratteli-Vershik representation $B'=\left( V,E,\preceq ' \right)$ (thus we only change the local orders) such that for all $n\geq 2$ and $u\in V_n$
the local orders of edges with range $u$ for $\preceq$ and $\preceq'$ differ at most $w_n$ times. 

\begin{coro}
\label{coro:necmodifying}
Let $(X,T)$ be a minimal Cantor system given
by a proper Bratteli-Vershik representation $B=\left( V,E,\preceq \right)$ and 
$\lambda=\exp(2i\pi \alpha)$ be a continuous eigenvalue of $(X,T)$. Let
$(\omega_n;n\geq 2)$ be a sequence of positive integers such that $\sum_{n\geq 2} \omega_{n+1} \vvert \alpha h_n \vvert $ converges.
Then, $\lambda$ is a continuous eigenvalue of any proper $(\omega_n;n\geq 2)$-order modification of $B$.
\end{coro}

\begin{proof}
Let $B'$ be a proper $(\omega_n;n\geq 2)$-order modification of $B$.
Let $s'_n$ and $s_n$ be the respective suffix vectors for each $n\geq 1$.
Then, $s_n = s'_n + \Delta_{n+1}$, with $|\Delta_{n+1}| \leq \omega_{n+1}$, and 
\begin{align*}
 \vvert\langle  s'_n(x) ,\alpha h_n  \rangle\vvert & \leq \vvert\langle  s_n(x) ,\alpha h_n  \rangle \vvert +  \vvert\langle  \Delta_{n+1} ,\alpha h_n  \rangle \vvert 
   \leq \vvert\langle  s_n(x) ,\alpha h_n  \rangle \vvert +  \omega_{n+1}\vvert  \alpha h_n  \vvert  .
\end{align*}

Since from Theorem \ref{theo:nec-suff-cont} one has $\sum_{n\ge 1}  \max_{x\in X} \vvert\langle  s_n(x) ,\alpha h_n  \rangle\vvert  <+\infty$ and by hypothesis $\sum_{n\geq 2} \omega_{n+1} \vvert \alpha h_n \vvert $ converges, then $\sum_{n\ge 1}  \max_{x\in X} \vvert\langle  s'_n(x) ,\alpha h_n  \rangle\vvert  <+\infty$. The result follows from Theorem \ref{theo:nec-suff-cont}.
\end{proof}

Before continuing the discussion let us recall the notions of orbit and strong orbit equivalence. Two dynamical systems $(X,T)$ and $(Y,S)$ are {\em orbit equivalent} if there exists a homeomorphism $\phi : X \to Y$ sending orbits to orbits: for all $x\in X$, 
$$
\phi \left(\{ T^n x \mid  n\in \mathbb{Z} \} \right) = \{ S^n \phi (x) \mid  n\in \mathbb{Z} \}.
$$
This induces the existence of maps $\vartheta : X\to \mathbb{Z}$ and $\kappa : X \to \mathbb{Z}$ satisfying for all $x\in X$,
$
\phi \circ T (x) = S^{\vartheta (x)}\circ \phi (x) \hbox{ and } \phi \circ T^{\kappa (x)} (x) = S\circ \phi (x).
$
When $\vartheta$ and $\kappa$ have both at most one point of discontinuity we say that $(X,T)$ and $(Y,S)$ are {\em strong orbit equivalent}. It is important to mention that orbit and strong orbit equivalence of two minimal Cantor systems essentially rely on the combinatorial information of the Bratteli-Vershik representations of such systems. In particular, two proper Bratteli-Vershik representations differing only on the local orders are strong orbit equivalent. 

In the context of our discussion, seminal papers \cite{gps} and \cite{ormes} show that continuous eigenvalues that are roots of the unity are preserved by strong orbit equivalence (this is also clear from Corollary \ref{coro:rootsofunity}). This is not the case of irrational continuous eigenvalues. In the next result we 
use Theorem \ref{theo:nec-suff-cont} to prove that Bratteli-Vershik representations of minimal Cantor systems (that are not topologically weakly mixing) can be modified in order to loose all their irrational continuous eigenvalues. At the same time, this can be done keeping measurable eigenvalues and being strong orbit equivalent with the original system. This result resembles the representation results of Ormes in \cite{ormes} that is used to prove that strong orbit equivalence of minimal Cantor systems is compatible with any group of measurable eigenvalues up to the roots of unity that are continuous eigenvalues.
The difference here is that the resulting system is not only strong orbit equivalent with the original one but it is obtained after precise contractions and  modifications of the local orders. In particular, both systems have the same topological rank which is not clear from \cite{ormes}.

For a deeper discussion and recent results on the relation of continuous eigenvalues with orbit equivalence see \cite{Cortez&Durand&Petite:2014}.

\begin{coro}
\label{coro:erasingcontinuous}
Let $(X,T)$ be a minimal Cantor system given
by a proper Bratteli-Vershik representation $B=\left( V,E,\preceq \right)$. 
Let $\mu$ be an ergodic probability measure.
There exists a telescoping of $B$ together with a change of local orders such that the resulting system has no irrational continuous eigenvalues and shares the same eigenvalues for $\mu$ with the original system. In particular, if the original system has no roots of the unity as continuous eigenvalues then the resulting system is topologically weakly mixing.  
\end{coro}
\begin{proof}
Let $(X,T)$ be the system given by the proper Brattelli-Vershik representation $B$ and let $\mu$ be an ergodic measure. As usual, to simplify we write $V_n=\{1,\ldots, d_n\}$ for all $n\geq 1$ and we assume that all maximal edges for the local orders start at vertex $d_n$ at each level.  

Let $(\ell_n; n\geq 0)$ be a strictly increasing sequence in $\NN$ with $l_0=0$ and $l_1=1$. We will fix it progressively by a recursive procedure. We contract the diagram $B$ associated to $(X,T)$ at levels $(\ell_{2n-1}; n\geq 1)$ to get a new representation $B'=(V',E',\preceq')$. Clearly, $E'_n=E_{\ell_{2n-3},\ell_{2n-1}}$ for all $n\geq 2$, so the combinatorial objects (matrices, height vectors, suffix vectors, etc.) and even measure vector associated to level $n$ of $B'$ correspond to those associated to the concatenation of levels $\ell_{2n-2}$ and $\ell_{2n-1}$ of $B$. The first levels coincide. 

For all $n\geq 2$ we modify the local orders of the edges in $E'_n$ by the following procedure. This kind of construction appears in the context of tiling systems in \cite{sadunpriebe}. For each $v \in V'_n=V_{\ell_{2n-1}}$ let $e_v \in E_{\ell_{2n-2},\ell_{2n-1}}$ be the maximal edge going from $d_{\ell_{2n-2}}$ to $v$. We order the paths in $E'_n$ finishing in vertex $v$ that passes through $e_v$ from left to right, {\it i.e.}, the order considers first edges starting in vertex $1$ and then follows with those starting in vertex $2$, then vertex $3$, etc., until arriving to edges starting in vertex $d_{\ell_{2n-3}}$. We keep the local order in the other edges. Call the new Bratteli-Vershik system $B''=(V'',E'',\preceq'')$ and the associated system $(Y,S)$. Since the combinatorial objects of diagram $B''$ are constructed from a telescoping of $B$, then $\mu$ is also an ergodic measure for $(Y,S)$. 

Let $(\alpha_n;n\in \NN)$ be a sequence containing infinitely many times all irrational real numbers such that $\lambda_n=\exp(2i\pi \alpha_n)$ verify the necessary conditions to be continuous eigenvalues of $(X,T)$ given by \eqref{eq:eta}. Since $B''$ is obtained from a telescoping of $B$ these real numbers are  also candidates to be continuous eigenvalues $(Y,S)$. But notice that some of the $\lambda_n$'s could not be continuous eigenvalues. We include all of them to be sure that our modifications will not add continuous eigenvalues. 
The construction of the sequence is possible since there are countable many of such real numbers. Indeed, 
since $\alpha_n$ verifies \eqref{eq:eta} for all $n\geq 1$, then for all $m \geq 1$,
$$\alpha_n h_m = \eta^{(n)}_m + \nu^{(n)}_m \hbox{ and } \eta^{(n)}_m \xrightarrow[m\to +\infty]{}0,$$ where $\nu^{(n)}_m$ is an integer vector. Since $\alpha_n$ is irrational, for any large enough $m$ we have that $\eta^{(n)}_m\not = 0$. Moreover, by \eqref{eq:eta2} $\alpha_n= \nu^{(n)}_m \cdot \mu_m$ for some $m\geq 1$. This proves that we have countably many possible values for the $\alpha_n$'s. 

First we prove that if the diagram has been contracted enough then all $\lambda_n$'s that are continuous eigenvalues for $(X,T)$ are also measurable eigenvalues of $(Y,S)$ for $\mu$. To start we can assume the diagram $B$ was contracted in such a way that 
$h_{\ell_{2n-1}}(v) \geq n$ for any $v \in V_{\ell_{2n-1}}$ and  $n\geq 2$.
\medskip

For $n\geq 2$ let $D_{n}$ be the set of points in $Y$ passing through $e_v$ for some $v \in V_{\ell_{2n-1}}$. We have, 
$$\mu(D_{n})=\sum_{v \in V_{\ell_{2n-1}}} \mu_{\ell_{2n-1}}(v) \ 
h_{\ell_{2n-2}}(d_{\ell_{2n-2}}).
$$  
But, $\sum_{v \in V_{\ell_{2n-1}}} \mu_{\ell_{2n-1}}(v)$ goes to zero as $n\to +\infty$. If not, there is a positive number $a$ such that for infinitely many $n \in \NN$ we have 
$1=h_{\ell_{2n-1}} \cdot \mu_{\ell_{2n-1}} \geq n a >0,$ 
which is a contradiction. Then, for all $n\geq 2$, if $\ell_{2n-2}$ is fixed, one can choose $\ell_{2n-1}$ to get 
$ \sum_{v \in V_{\ell_{2n-1}}} \mu_{\ell_{2n-1}}(v) \ 
h_{\ell_{2n-2}}(d_{\ell_{2n-2}}) \leq \varepsilon_{n},$ for a sequence of $\varepsilon_n$'s such that $\sum_{n\geq 2} \varepsilon_n < +\infty$. This choice implies that 
$\mu(\limsup_{n\to +\infty}D_{n})=0$.
Then, if $x\not \in \limsup_{n\to +\infty}D_{n}$, from a level $n(x) \geq 2$ it does not pass by any $e_v$ defined before. This means that the suffix vectors associated to $x$ in $B''$ differ in finitely many levels with those associated to $x$ in $B$. This implies that $\lambda_n^{r_m(x)}$ converges as $m\to +\infty$ in $B''$ for any $n\geq 1$. Then, by Theorem 7 in \cite{necesariasuficiente} (that we recall at the beginning of next section), $\lambda_n$ is a measurable eigenvalue associated to $\mu$.   


For technical reasons, later we will also need to have $\eta^{(n)}_{\ell_{2n-3}} < 1/4$ for all $n\geq 3$, so contracting a little more we can get this condition too.  


Now let us prove that a good choice of levels $(\ell_{2n-2};n\geq 2)$ allows to prove that the $\lambda_n$'s are not continuous eigenvalues of the modified system. Notice that we still have a lot of flexibility to choose recursively the values of $(\ell_{2n-2};n\geq 2)$ without changing previous results. 


We use the criterium given by Theorem \ref{theo:nec-suff-cont}. 
Fix $n\geq 2$. A careful inspection of the modified local order shows that for any $v\in V_{\ell_{2n-1}}$  
$$s''_{n-1}(v,t)=(0,\ldots,0,t) \text{ are suffix vectors in $S''_{n-1,n}=S_{\ell_{2n-3},\ell_{2n-1}}(d_{\ell_{2n-3}},v)$},$$
where $t\in \{0,\ldots,P_{\ell_{2n-3},\ell_{2n-2}}(d_{\ell_{2n-3}},d_{\ell_{2n-2}})-1\}$.
Take $\ell_{2n-2}$ large enough such that 
$$\frac{1}{\eta^{(n)}_{\ell_{2n-3}}(d_{\ell_{2n-3}})} < P_{\ell_{2n-3},\ell_{2n-2}}(d_{\ell_{2n-3}},d_{\ell_{2n-2}})-1.$$
Notice that the choice of the values of $\ell_{2n-2}$ and $\ell_{2n-1}$ for all $n \geq 2$ can be done recursively to get all desired properties. 

Then, for $t=\ds \left \lfloor \frac{1}{2 \eta^{(n)}_{\ell_{2n-3}}(d_{\ell_{2n-3}})}\right \rfloor +1$ we have  
$$
1/4 < \vvert < s_{n-1}''(v,t), \alpha_n h_{\ell_{2n-3}} > \vvert = 
\vvert t \cdot \eta^{(n)}_{\ell_{2n-3}}(d_{\ell_{2n-3}})  \vvert < 3/4,
$$
for all large enough $n$. 

Since each value in the sequence of $\alpha_n$'s appears infinitely many times, then the series in the condition of Theorem \ref{theo:nec-suff-cont} cannot converge for any $\alpha_n$, and thus $\lambda_n$ is not a continuous eigenvalues of $(Y,S)$ for all $n\in \NN$. 
\end{proof}

In Corollary \ref{coro:erasingcontinuous}, modifying the local orders we are able to eliminate all the irrational continuous eigenvalues of a given proper Bratteli-Vershik system. It is natural to ask whether it is possible, starting from a minimal Cantor system $(X,T)$ with $G$ as its group of continuous eigenvalues, to realise any subgroups $G'$ as the group of eigenvalues of some minimal Cantor system $(Y,S)$ that is strong orbit equivalent with $(X,T)$, or, strongly, to obtain $(Y,S)$ from $(X,T)$ contracting and modifying the local orders of a Bratteli-Vershik representation as we did in the last corollary. 

Proposition 25 in \cite{Cortez&Durand&Petite:2014} (see also \cite{Itza-Ortiz:2007}) asserts in specific cases that it is possible to realise some subgroups, but it is not completely clear which one can be realised. This is a purely theoretical result and we have no idea how to obtain the desired system with the prescribed subgroup of continuous eigenvalues just by modifying the local orders and telescoping. 
Moreover, there are strong obstructions for this kind of realisations and it is also shown in \cite{Cortez&Durand&Petite:2014} that this is not possible in general. For example, it is well known that the group of eigenvalues of a Sturmian subshift is $G = \{ \exp (2i\pi  \beta ); \beta \in \mathbb{Z} +\alpha \mathbb{Z}  \}$, where $\alpha$ is the angle of the rotation associated to the Sturmian system. The main result in \cite{Cortez&Durand&Petite:2014} shows that the only other subgroup of $G$ that can be realised within the strong orbit equivalence class of the Sturmian system is the other trivial subgroup. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Measurable eigenvalues of finite rank minimal Cantor systems}
\label{sec:measurable}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let us start with an abstract condition to be measurable eigenvalue.  

\begin{theo}[\cite{necesariasuficiente}, Theorem 7]\label{theo:cns_old_rho}
Let $(X,T)$ be a minimal Cantor system given by a proper  Bratteli-Vershik representation. Let  $\mu$ be an ergodic probability measure. Then $\lambda=\exp(2i \pi \alpha)$ is an eigenvalue of $(X,T)$  for $\mu$ if and only if there exists a sequence of functions
$(\rho_n:V_n\to \RR \talque n\geq 1)$
such that a subsequence of $\left(\alpha(r_n+\rho_n\circ\tau_n) \talque n\geq 1\right)$ converges $(\mod \ZZ)$ $\mu$-almost everywhere in $X$.
\end{theo}

The main problem in this condition, limiting its application, is the understanding of maps $\rho_n$. 
In this section we succeed to provide a necessary and sufficient condition to be a measurable eigenvalue of a finite rank minimal Cantor system that does not depend on the existence of the maps $\rho_n$ and, moreover, this condition gives insights into the way such maps can be constructed (Theorem \ref{theo:cns_sinrho}). It is based only on the combinatorial structure of Brattelli-Vershik representations of finite rank minimal Cantor systems. The drawback is that we impose some uniform convergence. In Section \ref{examples} we develop the Toeplitz case to show a nice application of the condition. 

Another necessary and sufficient condition valid for systems of exact rank ({\it i.e.}, $I_\mu=\{1,\ldots,d\}$) is presented in Theorem \ref{theo:cnsvpmedible}. It is formulated in the form of a convergence of a series, but again the term of the sum depends on the existence of auxiliary functions $\rho_n$ that we do not control. We include this condition since it follows previous works on the subject in the linearly recurrent case and can be an option to avoid the uniform convergence of the first one when such auxiliary functions are clear. 

In Section \ref{sec:modifying} we have discussed the strong relations between the group of continuous eigenvalues and the notion of strong orbit equivalence. In the case of measurable eigenvalues the situation is completely different. A first difference is that irrational measurable eigenvalues can not be interpreted in terms of dimension groups. Counterexamples can be constructed from results in \cite{ormes} (Theorem 6.1), generalizing Jewett-Krieger Theorem to strong orbit equivalence classes. In the same article, 
Ormes obtained a remarkable generalization of Dye's theorem \cite{Dye} involving strong orbit equivalence. From this result one can prove the existence of strong orbit equivalent minimal Cantor systems with prescribed group of measurable eigenvalues. Thus, for example, rational eigenvalues are not preserved under orbit equivalence. In \cite{Cortez&Durand&Petite:2014} is given an example showing that continuous rational eigenvalues are not even preserved.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Necessary and sufficient condition controlled by the ordering of the Bratteli-Vershik representation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $(X,T)$ be a minimal Cantor system given by a proper and clean Bratteli-Vershik representation of finite rank $d$ and let $\mu$ be an ergodic probability measure. We start with a classical analysis of an eigenfunction $f\in L^{2}(X,\mu)$ with $\abs{f}=1$, associated to some eigenvalue 
$\lambda$. Taking $n\geq 1$ we have

\begin{align*}
\EE(f|\T_n) &= \sum_{u\in V_n}\sum_{j=0}^{h_n(u)-1}{\bf 1}_{T^{-j}B_n(u)}\frac{1}{\mu_n(u)}\int_{T^{-j}B_n(u)}fd\mu \\
 &= \sum_{u\in V_n}\sum_{j=0}^{h_n(u)-1}{\bf 1}_{T^{-j}B_n(u)}\frac{1}{\mu_n(u)}\int_{B_n(u)}f\circ T^{-j}d\mu \\
 &= \sum_{u\in V_n}\sum_{j=0}^{h_n(u)-1}{\bf 1}_{T^{-j}B_n(u)}\frac{1}{\mu_n(u)}\int_{B_n(u)}\lambda^{-j}fd\mu.
\end{align*}

We define for each $n\geq 1$ and $u\in V_n$ the real numbers $c_n(u)$ and 
$\rho_n(u)$ in $[0,1]$ by 
\begin{equation}\label{eq:definicion_c_ro}
c_n(u)\lambda^{-\rho_n(u)}=\frac{1}{\mu_n(u)}\int_{B_n(u)}f d\mu  .
\end{equation}
Then we can write $\ds \EE(f|\T_n)(x) = c_n(\tau_n(x))\lambda^{-r_n(x)-\rho_n(\tau_n(x))}$, where we recall $r_n(x)$ is the entrance time of $x$ to $B_n(\tau_n(x))$.
\smallskip 

We have the following known property.
\begin{lemma}[\cite{rangofinito}, Lemma 17]\label{lemma:convergencia_c}
For each vertex $u \in\set{1,\ldots, d}$ 
$$
\mu(\tau_n=u)(1-c_n(u))\xrightarrow[n\to+\infty]{}0,
$$
and therefore for each $u\in I_{\mu}$, $c_n(u)\xrightarrow[n\to+\infty]{}1$.
\end{lemma}
Notice that we have identified $V_n$ with $\set{1,\ldots, d}$ for each $n\geq 1$.
\medskip

The following lemma will be useful to better understand our main result.
\medskip

\begin{lemma}\label{lemma:comportamiento_cuocientes}
Let $(X,T)$ be a minimal Cantor system given by a proper and clean Bratteli-Vershik representation of finite rank $d$, and let $\mu$ be an ergodic probability measure of the system. Then,
\begin{itemize}
\item[(1)] For $u\not\in I_{\mu}$ and $v\in I_{\mu}$
$$
\frac{h_m(u)}{h_n(v)}P_{m,n}(u,v)\xrightarrow[m\to+\infty]{}0
$$
uniformly for $n>m$. 
\item[(2)] For each $m\geq 1$, $u\in V_m$ and $v\in I_{\mu}$
$$
\frac{h_m(u)}{h_n(v)}P_{m,n}(u,v)\xrightarrow[n\to+\infty]{} \mu(\tau_m=u).
$$
\end{itemize}
\end{lemma}

\begin{proof}
(1) Recall $\delta_0 >0$ is the constant appearing in the definition of clean diagram, and take $u\not\in I_{\mu}$ and $v\in I_{\mu}$ (so $\mu(\tau_n=v) \geq \delta_0$ for every $n\geq 1$). To prove the statement it suffices to notice that the following inequality is true for every $n>m$:
\begin{align*}
\mu(\tau_m=u) &\geq \frac{h_m(u)}{h_n(v)}P_{m,n}(u,v)\mu(\tau_n=v) \geq \delta_0\frac{h_m(u)}{h_n(v)}P_{m,n}(u,v),
\end{align*}
and to make $m\to+\infty$.

Statement (2) can be proved using the same ideas in \cite[Lemma 11]{dfm}.
Recall that the measure of the set $B_m(u)$ is denoted by $\mu_m(u)$. Set $m\geq 1$, $u\in V_m$, $v\in I_\mu$ and $0< \varepsilon < \delta_0$. 
For $\mu$-almost every $x\in X$, the pointwise ergodic theorem and Egorov's theorem give us a set $A$ with $\mu(A)>1-\varepsilon$ and a positive integer $N_0$ such that for all $x\in A$ and $N\geq N_0$
\begin{equation}\label{eq:ergodico_egorov}
\abs{\frac{1}{N}\sum_{k=0}^{N-1} {\bf 1}_{B_m(u)}(T^{k}x) - \mu_m(u) }< \varepsilon.
\end{equation}

Let us take $n>m$ such that $h_n(v)>N_0$. We can find $j$, with $0\leq j\leq \left\lfloor\frac{\varepsilon h_n(v)}{\delta_0}\right\rfloor$, such that $A\cap T^{-h_n(v)-j+1}B_n(v)\neq\emptyset$. Indeed, since $\mu$ is invariant and $v\in I_{\mu}$ we have 
\begin{align*}
 &\mu\left(\bigcup_{j=0}^{\lfloor\varepsilon h_n(v)/\delta_0\rfloor}T^{-h_n(v)-j+1}B_n(v)\right) 
 = \left(\left\lfloor\frac{\varepsilon h_n(v)}{\delta_0}\right\rfloor + 1\right)\mu_n(v) > \frac{\varepsilon}{\delta_0}\mu(\tau_n=v)\geq \varepsilon.
\end{align*} 
\smallskip

Now, taking $x^{(n)}\in A\cap T^{-h_n(v)-j+1}B_n(v)$, relation \eqref{eq:ergodico_egorov} implies
\begin{equation}\label{eq:ergodico_egorov_hn}
\abs{\frac{1}{h_n(v)+j}\!\!\!\!\sum_{k=0}^{h_n(v)+j-1}\!\!\!\!\! {\bf 1}_{B_m(u)}(T^{k}x^{(n)}) - \mu_m(u) } < \varepsilon.
\end{equation}
Let us write
\begin{align*}
&\frac{1}{h_n(v)+j}\!\!\!\!\sum_{k=0}^{h_n(v)+j-1}\!\!\!\!\!{\bf 1}_{B_m(u)}(T^{k}x^{(n)}) \\
= & \frac{1}{h_n(v)+j}\sum_{k=0}^{j-1}{\bf 1}_{B_m(u)}(T^{k}x^{(n)})  + \frac{1}{h_n(v)}\!\!\!\!\sum_{k=j}^{h_n(v)+j-1}\!\!\!\!\!{\bf 1}_{B_m(u)}(T^{k}x^{(n)}) \\
 &- \frac{j}{h_n(v)(h_n(v)+j)}\!\!\!\!\sum_{k=j}^{h_n(v)+j-1}\!\!\!\!\!{\bf 1}_{B_m(u)}(T^{k}x^{(n)}).
\end{align*}
Notice that the modulus of the first and third terms on the right side are each bounded by
$$
\frac{j}{h_n(v)+j} < \frac{\varepsilon}{\delta_0}.
$$
Combining this with \eqref{eq:ergodico_egorov_hn} we obtain
\begin{equation}\label{eq:ergodico_egorov_final}
\abs{\frac{1}{h_n(v)}\!\!\!\!\sum_{k=j}^{h_n(v)+j-1}\!\!\!\!\!{\bf 1}_{B_m(u)}(T^{k}x^{(n)}) - \mu_m(u)} < \varepsilon\!\left(1+\frac{2}{\delta_0}\right).
\end{equation}
If we define $y^{(n)}=T^{h_n(v)+j-1}x^{(n)}\in B_n(v)$ (notice that $y^{(n)}$ depends on $n$ and $v$), relation \eqref{eq:ergodico_egorov_final} leads to
\begin{equation}\label{eq:ergodico_egorov_convergencia}
\frac{P_{m,n}(u,v)}{h_n(v)}=\frac{1}{h_n(v)}\!\!\sum_{k=0}^{h_n(v)-1}\!\!\!{\bf 1}_{B_m(u)}(T^{-k}y^{(n)})
\xrightarrow[n\to+\infty]{} \mu_m(u).
\end{equation}
Multiplying both sides of \eqref{eq:ergodico_egorov_convergencia} by $h_m(u)$ gives statement (2). 
\end{proof}

The next theorem is the main result of the section. 

\begin{theo}
\label{theo:cns_sinrho}
Let $(X,T)$ be a minimal Cantor system given by a proper and clean Bratteli-Vershik representation of finite rank $d$. Let  $\mu$ be an ergodic probability measure. 
Then $\lambda = \exp(2 i \pi \alpha)$ is an eigenvalue of $(X,T)$ for $\mu$
if and only if one of the following two equivalent conditions hold:
\begin{itemize}
\item[(1)] For all $v\in I_{\mu}$,
$$
\sum_{u\in I_{\mu}}\frac{h_m(u)}{h_n(v)}\abs{\sum_{s\in \sufijocbb{m}{n}{u}{v}}\lambda^{\la s, h_m\ra}}\xrightarrow[m\to+\infty]{} 1
$$
uniformly for $n>m$.
\vskip 0.5cm

\item[(2)] For all $u\in\set{1,\ldots,d}$ and $v\in I_{\mu}$,
$$
\frac{h_m(u)}{h_n(v)}\left[\matrizp{m}{n}{u}{v}-\abs{\sum_{s\in \sufijocbb{m}{n}{u}{v}}\lambda^{\la s, h_m\ra}}\right]\xrightarrow[m\to+\infty]{} 0
$$
uniformly for $n>m$
\end{itemize}
\end{theo}
Notice that in part (2) we are identifying $V_m$ with $\{1,\ldots,d\}$ for $m\geq 1$.

The proof of the complete statement has been divided into three parts.

%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Proof that (1) and (2) are equivalent}
%%%%%%%%%%%%%%%%%%%%%%%%%%
First, (2) implies (1) follows from $\sum_{u\in V_m} \frac{h_m(u)}{h_n(v)} \matrizp{m}{n}{u}{v}=1$ and part (1) of Lemma \ref{lemma:comportamiento_cuocientes}. 

To prove that (1) implies (2) we proceed by contradiction. Consider $\varepsilon >0$ 
and use (1) to get $M_0 \geq 1$ such that for all $n > m\geq M_0$ and all $v \in I_{\mu}$ we have 
\begin{align}\label{eq:blabla}
\sum_{u \in I_{\mu}} \frac{h_m(u)}{h_n(v)}\abs{\sum_{s\in \sufijocbb{m}{n}{u}{v}}\lambda^{\la s, h_m\ra}} > 1-\varepsilon \ . 
\end{align}
Now, assume that for some large $n>m$ and for some $u_0 \in V_m$ and $v_0 \in I_\mu$
we have that  
$$
\frac{h_m(u_0)}{h_n(v_0)}\left[\matrizp{m}{n}{u_0}{v_0}-\abs{\sum_{s\in \sufijocbb{m}{n}{u_0}{v_0}}\lambda^{\la s, h_m\ra}}\right] \geq \varepsilon \ . 
$$
Then, 
\begin{align*}
1 &= \sum_{u \in V_m} \frac{h_m(u)}{h_n(v_0)} \matrizp{m}{n}{u}{v_0} \\
  &\geq \varepsilon + \sum_{u \in V_m} \frac{h_m(u)}{h_n(v_0)}\abs{\sum_{s\in \sufijocbb{m}{n}{u}{v_0}}\lambda^{\la s, h_m\ra}} \\
  &\geq \varepsilon + \sum_{u \in I_\mu } \frac{h_m(u)}{h_n(v_0)}\abs{\sum_{s\in \sufijocbb{m}{n}{u}{v_0}}\lambda^{\la s, h_m\ra}} > 1 \ , \\
\end{align*}
where in the last inequality we have used \eqref{eq:blabla}.
This is a contradiction and (2) follows. 


\subsubsection{Proof of the necessity of the conditions}
We start noticing that the atom $B_{m}(u)$ of the associated Kakutani-Rohlin partition $\mathcal{P}_m$, for $m\geq 1$ and $u \in V_m$, can be written as a disjoint union of elements of $\mathcal{P}_n$ for $n>m$ in the following way
\begin{equation}\label{eq:union_disjunta}
B_{m}(u) = \bigcup_{v\in V_n} \bigcup_{s\in \sufijocbb{m}{n}{u}{v}} T^{-\la s,h_m\ra}B_n(v).
\end{equation}

Applying $\mu$ on both sides of \eqref{eq:union_disjunta} and introducing heights to get measures of towers we have
\begin{equation}\label{eq:medidas_torres}
\mu(\tau_m=u)=\sum_{v \in V_n}\frac{h_m(u)}{h_n(v)}P_{m,n}(u,v)\mu(\tau_n=v).
\end{equation}

On the other side, we integrate a fixed eigenfunction $f$ of modulus $1$ associated to $\lambda$ over $B_m(u)$. 
We use equality \eqref{eq:union_disjunta} to obtain
\begin{align}
\int_{B_m(u)}fd\mu &= \sum_{v \in V_n}\sum_{s\in \sufijocbb{m}{n}{u}{v}}\int_{B_n(v)}f\circ T^{-\la s,h_m\ra}d\mu \nonumber \\
  &= \sum_{v\in V_n}\left(\sum_{s\in \sufijocbb{m}{n}{u}{v}}\!\!\!\lambda^{-\la s,h_m\ra}\right)\int_{B_n(v)}f d\mu \ ,  \label{eq:integrando_funcionpropia}
\end{align}

and then, applying \eqref{eq:definicion_c_ro} on \eqref{eq:integrando_funcionpropia} and multiplying both sides by $h_m(u)$ we get
\begin{align}
& \mu(\tau_m=u)c_m(u)\lambda^{-\rho_m(u)} \nonumber \\
&\qquad\qquad = \sum_{v\in V_n}\frac{h_m(u)}{h_n(v)}\left(\sum_{s\in \sufijocbb{m}{n}{u}{v}}\!\!\!\lambda^{-\la s,h_m\ra}\right)
\mu(\tau_n=v)c_n(v)\lambda^{-\rho_n(v)}. \label{eq:medidas_torres_mas_c_rho_lambda}
\end{align}

With these two very similar equations, \eqref{eq:medidas_torres} and \eqref{eq:medidas_torres_mas_c_rho_lambda}, we can conclude the ``{}necessity part''{} of the proof in the following way.
First, we have the inequalities
\begin{equation}
|c_n (v) |\leq 1 \hbox{ and } \bigg|\!\sum_{s\in \sufijocbb{m}{n}{u}{v}}\!\!\!\lambda^{-\la s,h_m\ra}\bigg|\leq P_{m,n}(u,v),
\end{equation}
so we take the absolute value on both sides of \eqref{eq:medidas_torres_mas_c_rho_lambda} to obtain
\begin{align*}
\mu(\tau_m=u)c_m(u) &\leq \sum_{v\in V_n}\frac{h_m(u)}{h_n(v)}\bigg|\!\sum_{s\in \sufijocbb{m}{n}{u}{v}}\!\!\!\lambda^{-\la s,h_m\ra}\bigg|\mu(\tau_n=v) \\
  &\leq \sum_{v \in V_n}\frac{h_m(u)}{h_n(v)}P_{m,n}(u,v)\mu(\tau_n=v) \\
  &= \mu(\tau_m=u).
\end{align*}
Then applying Lemma \ref{lemma:convergencia_c} we see that
\begin{equation}
\sum_{v \in V_n}\frac{h_m(u)}{h_n(v)}\left[P_{m,n}(u,v)-\bigg|\!\sum_{s\in \sufijocbb{m}{n}{u}{v}}\!\!\!\lambda^{-\la s,h_m\ra}\bigg|\right]\mu(\tau_n=v)\xrightarrow[m\to+\infty]{}0
\end{equation}
uniformly for $n>m$.

Finally, for any $v\in I_{\mu}$ we get
$$
\frac{h_m(u)}{h_n(v)}\left[P_{m,n}(u,v)-\bigg|\!\sum_{s\in \sufijocbb{m}{n}{u}{v}}\!\!\!\lambda^{-\la s,h_m\ra}\bigg|\right]\xrightarrow[m\to+\infty]{}0
$$
uniformly for $n>m$, which is condition (2) of Theorem \ref{theo:cns_sinrho} (recall that $u$ is an arbitrary vertex in $V_m$).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Proof of the sufficiency of the conditions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now we assume $$\frac{h_m(u)}{h_n(v)}\left[\matrizp{m}{n}{u}{v}-\bigg|\!\!\sum_{s\in\sufijocbb{m}{n}{u}{v}}\!\!\!\lambda^{\la s,h_m\ra}\bigg|\right] \xrightarrow[m\to+\infty]{}0$$ uniformly in $n>m\geq 1$ for $u \in V_m$ and $v \in I_\mu$.

We start with the following lemma, which will allow us to handle the sum of powers of $\lambda$ that appear in the (equivalent) conditions of Theorem \ref{theo:cns_sinrho}.
\begin{lemma}[Geometric Lemma]\label{lemma:geometrico}
Let $z_k=\exp(2i \pi \alpha_k)$ with $\alpha_k\in [0,1)$ for $k=1,\ldots, N$ be $N$ complex numbers, where $N>1$, and let $\varepsilon\leq 1$ and $\gamma\leq 1/2$ be positive real numbers.

Suppose that $\ds \Big |\sum_{k=1}^{N}z_k\Big |>(1-\varepsilon)N$. Then there exists $1\leq \ell\leq N$ such that
$$
\#\set{\alpha_k \talque 1\leq k\leq N \y \vvert \alpha_k - \alpha_{\ell} \vvert \geq \gamma} <\frac{2N\varepsilon}{1-\cos(2\pi\gamma)}.
$$
\end{lemma}
\begin{proof}
Set $S=z_1 + z_2 + \cdots + z_N$. It is easy to check that $S$ satisfies
$$
|S|^{2} = N + 2\!\!\!\sum_{1\leq i<j\leq N}\!\!\cos(2\pi\vvert \alpha_i - \alpha_j\vvert).
$$
Then, from $|S|>(1-\varepsilon)N $ one gets $1-\frac{|S|^{2}}{N^{2}} < 2\varepsilon $, and thus 
\begin{align}
2\!\!\!\sum_{1\leq i<j\leq N}\!\!\cos(2\pi\vvert \alpha_i - \alpha_j\vvert) > N(N-1) -2N^{2}\varepsilon. \label{eq:geom1}
\end{align}
On the other hand, if we define
$$
K=\set{(\alpha_i, \alpha_j) \talque 1\leq i,j\leq N \y \vvert \alpha_i - \alpha_j \vvert \geq \gamma},
$$
then
\begin{align}
& 2\!\!\!\sum_{1\leq i<j\leq N}\!\!\cos(2\pi\vvert \alpha_i - \alpha_j\vvert) \nonumber \\
=&\sum_{1\leq i\neq j\leq N}\!\!\cos(2\pi\vvert \alpha_i - \alpha_j\vvert) \nonumber \\
=& \sum_{\substack{(\alpha_i,\alpha_j)\in K \\ 1\leq i,j\leq N}}\!\!\cos(2\pi\vvert \alpha_i - \alpha_j\vvert) + \sum_{\substack{(\alpha_i,\alpha_j)\not\in K \\ 1\leq i\neq j\leq N}}\!\!\cos(2\pi\vvert \alpha_i - \alpha_j\vvert) \nonumber \\
\leq &\# K\cos(2\pi\gamma) + N(N-1) - \#K.
\end{align}

Combining this last inequality with \eqref{eq:geom1} we deduce that
$$
\# K < \frac{2N^{2}\varepsilon}{1-\cos(2\pi\gamma)}.
$$
So it should exist $1\leq \ell\leq N$ such that
$$
\#\set{\alpha_k \talque 1\leq k\leq N \y \vvert \alpha_k - \alpha_{\ell} \vvert \geq \gamma} <\frac{2N\varepsilon}{1-\cos(2\pi\gamma)}.
$$
\end{proof}

From now on we will make, by telescoping the associated Bratteli-Vershik diagram if necessary, the following assumptions: 
\begin{itemize}
\item[(1)] For all $u\not\in I_{\mu}$,
\begin{equation}\label{eq:torres_despreciables_medidas_sumables}
\sum_{n=1}^{\infty}\mu(\tau_n=u) < \infty
\end{equation}
(not only $\mu(\tau_n=u)\xrightarrow[n\to+\infty]{}0$ as in the definition of clean diagram).
\item[(2)] For all $n>m\geq 1$ and $u,v\in I_{\mu}$,
\begin{equation}\label{eq:cota_inf_cuocientes}
\frac{h_m(u)}{h_n(v)}\matrizp{m}{n}{u}{v} > \frac{\delta_0}{2},
\end{equation} 
where $\delta_0$ is the constant from the definition of clean diagram. It is clear that Lemma \ref{lemma:comportamiento_cuocientes}(2) allows us to assume this.
\end{itemize}




The proof of the ``{}sufficiency part''{} of Theorem \ref{theo:cns_sinrho} consists in constructing functions $\rho_n:V_n\to \RR$, with $n\geq 1$, as in the formulation of Theorem \ref{theo:cns_old_rho}, and proving the convergence associated with them. In order to accomplish this, we will break the proof below into several steps. 
\medskip

The first step consists in constructing with the help of Lemma \ref{lemma:geometrico} some useful sequences $(\Lambda_{m,n}(u,v) \talque n>m\geq 1 \y u,v\in I_{\mu})$ and $(\mathcal{D}_{m,n} \talque n>m\geq 1)$, of integers and measure-decreasing sets respectively, and show some relevant properties.


For $n>m\geq 1$ and $u,v\in I_{\mu}$ write
\begin{equation}\label{eq:def_Delta}
\Delta_{m,n}(u,v) = \frac{h_m(u)}{h_n(v)}\left[\matrizp{m}{n}{u}{v}-\bigg|\!\!\sum_{s\in\sufijocbb{m}{n}{u}{v}}\!\!\!\lambda^{\la s,h_m\ra}\bigg|\right].
\end{equation}

Using \eqref{eq:cota_inf_cuocientes} we can see that
$$
\Delta_{m,n}(u,v) > \frac{\delta_0}{2}\left[1-\frac{1}{\matrizp{m}{n}{u}{v}}\bigg|\!\!\sum_{s\in\sufijocbb{m}{n}{u}{v}}\!\!\!\lambda^{\la s,h_m\ra}\bigg|\right],
$$
and therefore
\begin{equation}\label{eq:desigualdad_lambda_Delta_P}
\bigg|\!\!\sum_{s\in\sufijocbb{m}{n}{u}{v}}\!\!\!\lambda^{\la s,h_m\ra}\bigg|>\left(1-\frac{2\Delta_{m,n}(u,v)}{\delta_0}\right)\!\matrizp{m}{n}{u}{v}.
\end{equation}

Now, consider
\begin{equation}\label{eq:def_gamma_mn}
\gamma=\gamma_{m,n}(u,v) = \frac{1}{2\pi}\arccos\left(1-\sqrt{\Delta_{m,n}(u,v)}\right),
\ \varepsilon=\varepsilon_{m,n}(u,v) =\frac{2\Delta_{m,n}(u,v)}{\delta_0}
\end{equation}
and notice that with this choice of $\gamma$ and $\varepsilon$, taking  large enough $n > m \geq 1$ and inequality \eqref{eq:desigualdad_lambda_Delta_P}, the hypothesis of Lemma \ref{lemma:geometrico} holds for the complex numbers $\lambda^{\la s,h_m\ra}$ (recall that we are assuming $\Delta_{m,n}(u,v)\xrightarrow[m\to+\infty]{} 0$ uniformly for $n>m\geq 1$, for all $u,v\in I_{\mu}$). We deduce that there exists 
$s_0\in\sufijocbb{m}{n}{u}{v}$ such that
\begin{align}
& \#\set{s\in\sufijocbb{m}{n}{u}{v} \talque \vvert\alpha\la s,h_m \ra - \alpha\la s_0,h_m \ra\vvert \geq \gamma_{m,n}(u,v)} \nonumber \\
< &  \frac{4\matrizp{m}{n}{u}{v}}{\delta_0}\sqrt{\Delta_{m,n}(u,v)}. \label{eq:lema_geom_aplicado}
\end{align}
Notice that $s_0$ depends on $m,n,u$ and $v$. Also, $s_0$ is chosen arbitrarily for ``{}not large enough''{} values of $n>m \ge 1$.

For $n>m\geq 1$ and $u,v\in I_{\mu}$, write $\Lambda_{m,n}(u,v)=\la s_0,h_m\ra$ and define $\mathcal{D}_{m,n}$ as the set of points $x \in X$ such that 
$\tau_m (x), \tau_n(x)\in I_\mu$ and
$$
\vvert\alpha\la s_{m,n}(x),h_m\ra - \alpha\Lambda_{m,n}(\tau_m(x),\tau_n(x))\vvert \geq \gamma_{m,n}(\tau_m(x),\tau_n(x)).
$$

\begin{lemma}\label{lemma:mu_D_converge_a_cero}
$\ds\mu(\mathcal{D}_{m,n})\xrightarrow[m\to+\infty]{}0$ uniformly for $n>m\geq 1$.
\end{lemma}

\begin{proof}
For $n>m\geq 1 $ the measure of $\mathcal{D}_{m,n}$ can be written
$$
\sum_{u\in I_{\mu}}\sum_{v\in I_{\mu}}\mu(\vvert\alpha\la s_{m,n}, h_m \ra - \alpha\Lambda_{m,n}(u,v)\vvert
\geq \gamma_{m,n}(u,v), \tau_m = u, \tau_n=v).
$$
Assuming $m$ is large enough, from inequality \eqref{eq:lema_geom_aplicado} we obtain
\begin{align}
\mu(\mathcal{D}_{m,n})  &< \sum_{u\in I_{\mu}}\sum_{v\in I_{\mu}}\frac{4\matrizp{m}{n}{u}{v}}{\delta_0}\sqrt{\Delta_{m,n}(u,v)}\mu_{n}(v)h_m(u) \nonumber\\
 &= \sum_{u\in I_{\mu}}\sum_{v\in I_{\mu}}\frac{4h_m(u)\matrizp{m}{n}{u}{v}}{\delta_0 h_n(v)}\sqrt{\Delta_{m,n}(u,v)}\mu(\tau_n=v) \nonumber \\
 &\label{eq:cota_Dmn} \leq \sum_{u\in I_{\mu}}\sum_{v\in I_{\mu}}\frac{4\sqrt{\Delta_{m,n}(u,v)}}{\delta_0}\mu(\tau_n=v).
\end{align}
The lemma follows since, by hypothesis, the right hand side goes to zero as desired.
\end{proof}

\begin{lemma}[Quasi-Additivity of $\alpha\Lambda_{m,n}(u,v)$]\label{lemma:cuasi_aditividad_Lambda}
For $n>m>\ell \geq 1$ large enough and $u,v,w \in I_{\mu}$ we have
\begin{align*}
\vvert\alpha\Lambda_{\ell,m}(u,v) + \alpha\Lambda_{m,n}(v,w) - \alpha\Lambda_{\ell,n}(u,w)\vvert   < \gamma_{\ell,m}(u,v) + \gamma_{m,n}(v,w) + \gamma_{\ell,n}(u,w).
\end{align*}
\end{lemma}

\begin{proof}
Fix $u,v,w \in I_{\mu}$. For $\ell < m <n$ write
\begin{align*}
\mathcal{V}_{\ell,m,n} = & \set{\tau_{\ell}=u, \tau_{m}=v, \tau_{n}=w} \\
&\cap \set{\vvert\alpha\la s_{\ell, m}, h_{\ell}\ra - \alpha\Lambda_{\ell,m}(\tau_{\ell},\tau_m)\vvert < \gamma_{\ell,m}(u,v)} \\
&\cap \set{\vvert\alpha\la s_{m, n}, h_{m}\ra - \alpha\Lambda_{m, n}(\tau_{m},\tau_n)\vvert < \gamma_{m,n}(v,w)}. \\
\end{align*}
First we estimate the measure of this set. Since the maps $s_{\ell, m}$ and $s_{m, n}$ only depend on levels from $\ell$ to $m$ and $m$ to $n$ respectively,  we can see from the structure of the measure $\mu$ that $\mu(\mathcal{V}_{\ell,m,n})$ is equal to
\begin{align*}
 &\#\set{s\in\sufijocbb{\ell}{m}{u}{v} \talque \vvert\alpha\la s, h_{\ell}\ra - \alpha\Lambda_{\ell,m}(u,v)\vvert < \gamma_{\ell,m}(u,v)} \\
 &\qquad\times \#\set{s\in\sufijocbb{m}{n}{v}{w} \talque \vvert\alpha\la s, h_{m}\ra - \alpha\Lambda_{m,n}(v,w)\vvert < \gamma_{m, n}(v,w)} \\
 &\qquad\times\mu_n(w)h_{\ell}(u).
\end{align*}
For $\ell<m<n$ large enough, inequality \eqref{eq:lema_geom_aplicado} shows that the two set cardinalities involved in the above expression can be bounded below by $\matrizp{\ell}{m}{u}{v}/2$ and $\matrizp{m}{n}{v}{w}/2$ respectively. So, it exists $N_0$ such that, for $n>m>\ell>N_0$,
\begin{align}
\mu(\mathcal{V}_{\ell,m,n}) &\geq \frac{\matrizp{\ell}{m}{u}{v}\matrizp{m}{n}{v}{w}\mu_n(w)h_{\ell}(u)}{4} \nonumber \\
 &= \frac{\mu(\tau_n=w)}{4}\frac{h_{\ell}(u)}{h_{m}(v)}\matrizp{\ell}{m}{u}{v}\frac{h_{m}(v)}{h_{n}(w)}\matrizp{m}{n}{v}{w} \nonumber \\
 &\geq \frac{\delta_0^{3}}{16}. \nonumber
\end{align}
Also, $N_0$ can be chosen such that  $\mu(\mathcal{D}_{\ell,n}) < \delta_0^{3}/16$, as a consequence of Lemma \ref{lemma:mu_D_converge_a_cero}.

Now we proceed by contradiction. Suppose that the assertion of the lemma is false for the fixed $u,v,w\in I_{\mu}$. Then we will find positive integers $n>m>\ell>N_0$ such that
\begin{align*}
\vvert\alpha\Lambda_{\ell,m}(u,v) + \alpha\Lambda_{m,n}(v,w) - \alpha\Lambda_{\ell,n}(u,w)\vvert 
\geq \gamma_{\ell,m}(u,v) + \gamma_{m,n}(v,w) + \gamma_{\ell,n}(u,w).
\end{align*}
We claim that for these positive integers we have $\mathcal{V}_{\ell,m,n}\subseteq \mathcal{D}_{\ell,n}$. Indeed, for any $x_0\in \mathcal{V}_{\ell,m,n}$, using \eqref{eq:formulareturn_mn_suma}, we get
\begin{align*}
& \vvert \alpha \la s_{\ell,n}(x_0), h_{\ell} \ra - \alpha\Lambda_{\ell,n}(\tau_{\ell}(x_0),\tau_n(x_0))\vvert  \\
= & \vvert \alpha \la s_{\ell,n}(x_0), h_{\ell} \ra - \alpha\Lambda_{\ell,n}(u,w)\vvert \\
 \geq & \vvert\alpha\Lambda_{\ell,m}(u,v) + \alpha\Lambda_{m,n}(v,w) - \alpha\Lambda_{\ell,n}(u,w)\vvert \\
  & - \vvert\alpha\Lambda_{\ell,m}(u,v)-\alpha\la s_{\ell,m}(x_0), h_{\ell}\ra\vvert - \vvert\alpha\Lambda_{m,n}(v,w)-\alpha\la s_{m,n}(x_0), h_{m}\ra\vvert \\
  > & \gamma_{\ell,n}(u,w),
\end{align*}

{\it i.e.}, $x_0\in \mathcal{D}_{\ell,n}$.
The inclusion $\mathcal{V}_{\ell,m,n}\subseteq \mathcal{D}_{\ell,n}$ contradicts $\mu(\mathcal{D}_{\ell,n}) < \delta_0^{3}/16 \leq \mu(\mathcal{V}_{\ell,m,n})$. 


This proves the lemma noticing that we have only a finite number of different choices for $u,v,w\in I_{\mu}$.
\end{proof}

Our next task consists in defining a suitable set of full measure (the complement of a set $\mathcal{C}$ of null measure) such that we can handle the size of $\vvert \alpha\la s_{m,n}, h_m\ra - \alpha\Lambda_{m,n}(\tau_m,\tau_n)\vvert$ for any of their elements.
To do this, fix a decreasing sequence of positive real numbers $(\varepsilon_n \talque n\geq 1)$ such that $\sum_{n\geq 1} \varepsilon_n <\infty$. By \eqref{eq:def_gamma_mn} and the hypothesis, we have that for all $u,v\in I_{\mu}$, $\gamma_{m,n}(u,v) \xrightarrow[m\to+\infty]{} 0$ uniformly for $n>m$. 
Hence, we can fix an increasing sequence of positive integers $(m_k \talque k\geq 1)$ such that for every $n> m_k$, with $k\geq 1$, and $u,v\in I_{\mu}$
\begin{equation}\label{gamma_sumable}
\gamma_{m_k,n}(u,v) \leq \varepsilon_k \ .
\end{equation}

For $n>m$, set
$$
\mathcal{C}_{m,n} = \mathcal{D}_{m,n} \cup \set{\tau_m\not\in I_{\mu}} \cup \set{\tau_n\not\in I_{\mu}}.
$$
\begin{lemma}\label{lemma:medida_C_es_nula}
Let $(m_k \talque k\geq 1)$ be the above-mentioned sequence, and set $\ds\mathcal{C}=\limsup_{k\to+\infty}\mathcal{C}_{m_k,m_{k+1}}$. Then $\mu(\mathcal{C})=0$.
\end{lemma}
\begin{proof}
Notice that for $n>m$ large enough and every $u, v \in I_{\mu}$ we have
$$
\Delta_{m,n}(u,v) < \sqrt{\Delta_{m,n}(u,v)} < \gamma_{m,n}(u,v).
$$
Hence $\sqrt{\Delta_{m_k,m_{k+1}}(u,v)} < \varepsilon_k$ for every $k\geq 1$ and $u,v \in I_{\mu}$. Using the bound \eqref{eq:cota_Dmn}, it can be obtained the summability of $(\mu(\mathcal{D}_{m_k,m_{k+1}}) \talque k\geq 1)$. Therefore
$$
\mu(\mathcal{C}_{m_k,m_{k+1}}) < \infty
$$
(recall \eqref{eq:torres_despreciables_medidas_sumables}), and the lemma follows.
\end{proof}

Finally, we proceed to construct the sequence $(\rho_m\talque m\geq 1)$ which is part of the main goal of this proof. To this end, fix $v_0\in I_{\mu}$, and by means of a standard diagonalization process we can find $(n_i\talque i\geq 1)$ such that for all $m\geq 1$ and $u\in I_{\mu}$, the sequence $(\alpha\Lambda_{m,n_i}(u,v_0)\talque i\geq 1)$ is convergent.
Considering this, we define
$$
\rho_m(u) = \left\{
\begin{array}{ccc}
\ds \frac{1}{\alpha}\lim_{i\to+\infty}\alpha\Lambda_{m,n_i}(u,v_0)\; (\operatorname{mod}\; \ZZ) &{\rm for}& u\in I_{\mu} \\
0&{\rm for}& u\not\in I_{\mu}.
\end{array}
\right.
$$

By Theorem \ref{theo:cns_old_rho}, we will establish the ``{}sufficient part''{} of Theorem \ref{theo:cns_sinrho} if we prove the following lemma.
\begin{lemma}\label{lemma:convergencia_rho}
Let $(m_k \talque k\geq 1)$ and $\mathcal{C}$ be as in the formulation of Lemma \ref{lemma:medida_C_es_nula}. Then, for all $x\in X\setminus \mathcal{C}$, 
$$
\left(\alpha(r_{m_k}(x) + \rho_{m_k}(\tau_{m_k}(x))) \talque k\geq 1\right)
$$
converges (mod $\ZZ$).
\end{lemma}
\begin{proof}
Take $\varepsilon>0$ and $x\not\in\mathcal{C}$. It exists a positive integer $k_0$ such that $x\not\in\mathcal{C}_{m_k,m_{k+1}}$, for all $k\geq k_0$. Here and subsequently $u_k$ denotes the vertex $\tau_{m_k}(x)$. By definition of $\mathcal{C}_{m_k,m_{k+1}}$ we have that $u_k\in I_{\mu}$ and 
\begin{equation}\label{eq:desigualdad_miembros_conjunto_bueno}
\vvert \alpha\la s_{m_{k},m_{k+1}}(x), 	h_{m_k}\ra  - \alpha\Lambda_{m_k,m_{k+1}}(u_k,u_{k+1})\vvert < \gamma_{m_k,m_{k+1}}(u_k,u_{k+1}),
\end{equation}
for $k\geq k_0$.

The integer $k_0$ will be chosen large enough such that $\sum_{k=k_0}^{\infty}\varepsilon_k < \varepsilon/8$ (recall the sequence $(\varepsilon_k\talque k\geq 1)$ from the construction of sequence $(m_k\talque k\geq 1)$).

Now fix $\ell > k \geq k_0$. By definition of the $\rho_m$'s we can find an integer $j\geq 0$ such that simultaneously
\begin{align}
\vvert \alpha\Lambda_{m_k,j}(u_k,v_0) - \alpha\rho_{m_k}(u_k)\vvert &< \varepsilon/4, \; \; \; {\rm and}\nonumber \\
\vvert \alpha\Lambda_{m_{\ell},j}(u_{\ell},v_0) - \alpha\rho_{m_{\ell}}(u_{\ell})\vvert &< \label{eq:Lambda_cerca_rho}\varepsilon/4.
\end{align}

For $k\leq i < \ell$ define
\begin{align*}
\Theta_i &= \alpha\Lambda_{m_i,m_{i+1}}(u_i,u_{i+1}) + \alpha\Lambda_{m_{i+1},j}(u_{i+1},v_0) - \alpha\Lambda_{m_i,j}(u_i,v_0), \\
\Omega_i &= \alpha\la s_{m_i,m_{i+1}}(x), h_{m_i}\ra - \alpha\Lambda_{m_i,m_{i+1}}(u_i,u_{i+1}).
\end{align*}

From the quasi additivity stated in Lemma \ref{lemma:cuasi_aditividad_Lambda} and \eqref{eq:desigualdad_miembros_conjunto_bueno} we have, for $k\leq i < \ell$,
\begin{align}
\vvert\Theta_i\vvert & < \gamma_{m_i,m_{i+1}}(u_i,u_{i+1}) + \gamma_{m_{i+1},j}(u_{i+1},v_0) + \gamma_{m_i,j}(u_i,v_0) \nonumber \\
 & < \varepsilon_i + \varepsilon_{i+1} + \varepsilon_i < 3\varepsilon_i, \label{eq:desigualdad_Theta_i}\\
\vvert\Omega_i\vvert & < \gamma_{m_i,m_{i+1}}(u_i,u_{i+1}) < \varepsilon_i \label{eq:desigualdad_Omega_i}
\end{align}
(notice that we could have chosen $k_0$ large enough in order to apply Lemma \ref{lemma:cuasi_aditividad_Lambda}), and with the help of properties \eqref{eq:formulareturn_mn} and \eqref{eq:suma_sufijos} of suffix vectors, we deduce the identity
\begin{equation}\label{eq:identidad_Theta_Omega_i}
\sum_{i=k}^{\ell-1}\Theta_i + \Omega_i  = \alpha \left[r_{m_{\ell}}(x) - r_{m_{k}}(x)\right] + \alpha\Lambda_{m_{\ell},j}(u_{\ell},v_0) - \alpha\Lambda_{m_{k},j}(u_{k},v_0).
\end{equation}

Combining \eqref{eq:Lambda_cerca_rho}, \eqref{eq:desigualdad_Theta_i}, \eqref{eq:desigualdad_Omega_i} and \eqref{eq:identidad_Theta_Omega_i} gives
\begin{align*}
\vvert \alpha\left(r_{m_{\ell}}(x) + \rho_{m_{\ell}}(\tau_{m_{\ell}}(x))\right) - \alpha\left(r_{m_{k}}(x) + \rho_{m_{k}}(\tau_{m_{k}}(x))\right)\vvert &  \\
 & \hspace{-9em}< \frac{\varepsilon}{2} + \sum_{i=k}^{\ell-1} \vvert\Theta_i\vvert + \vvert\Omega_i\vvert \\
 & \hspace{-9em}< \frac{\varepsilon}{2} + \sum_{i=k}^{\ell-1} 4\varepsilon_i < \frac{\varepsilon}{2} + 4\sum_{i=k}^{\infty} \varepsilon_i < \varepsilon.
\end{align*}
Therefore, $(\alpha\left(r_{m_{k}}(x) + \rho_{m_{k}}(\tau_{m_{k}}(x))\right)\talque k\geq 1)$
is a Cauchy sequence. This proves the lemma and consequently, Theorem \ref{theo:cns_sinrho}.
\end{proof}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{A necessary and sufficient condition for the exact finite rank case}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The general necessary and sufficient condition for a complex number to be the eigenvalue of a linearly recurrent minimal Cantor system (we recall they are uniquely ergodic) proposed in \cite{necesariasuficiente} is described in the form of a convergence of a series. In the case of finite rank minimal Cantor systems a similar condition was proved to be necessary in \cite{rangofinito} making use of the existence of auxiliary functions $\rho_n:V_n \to \RR$ as those constructed in the proof just above. In this more general framework, this condition depends on the ergodic measures of the system, each ergodic measure defining its own group of eigenvalues.
One virtue of this condition is that it does not need to handle uniform convergences in relation to several levels in the Bratteli-Vershik representation. 
Nevertheless, the auxiliary functions could be difficult to compute as was observed in the proof of previous section. 

Here we prove that the necessary condition to be an eigenvalue given in \cite{rangofinito} (Proposition 18) is actually sufficient in the exact finite rank case (we think the exact finite rank hypothesis is not necessary). 
We recall systems with exact finite rank were defined in \cite{bkms}, they are uniquely ergodic and are defined asserting that $I_\mu = \{ 1, \ldots , d\}$ for the unique ergodic measure $\mu$.

\begin{theo} 
\label{theo:cnsvpmedible}
Let $(X,T)$ be a minimal Cantor system given by a proper and clean Bratteli-Vershik representation of exact finite rank $d$ for  the ergodic probability measure $\mu$ 
({\it i.e.}, $I_\mu=\{1,\ldots,d\}$). 
Then $\lambda=\exp(2i \pi \alpha)$ is an eigenvalue of $(X,T)$ for $\mu$ if and only if for every $n\geq 1$ there exist functions $\rho_{n}: V_n \to \RR$ such that
\begin{equation}
\label{eq:cnsvpmedible}
\sum_{n\geq 1}  \frac{1}{M_{n+1}(u_{n},u_{n+1})} \sum_{s\in S_{n}(u_{n}, u_{n+1})} 
\left |\lambda^{\langle s,h_{n} \rangle-\rho_{n+1}(u_{n+1})+\rho_{n}(u_{n})}-1 \right |^{2} < +\infty
\end{equation}
for any sequence $(u_{n}; n\geq 1)$ in $I_{\mu}$.
\end{theo}
\vskip 0.5cm

To prove condition \eqref{eq:cnsvpmedible} is sufficient for $\lambda$ to be an eigenvalue of $(X,T)$ for $\mu$ we follow the same strategy developed in \cite{necesariasuficiente}. The proof will be a consequence of several steps developed in the next subsections where we follow notations in Theorem \ref{theo:cnsvpmedible} and we assume \eqref{eq:cnsvpmedible} holds.

\subsubsection{An alternative formulation} Let 
us define $\theta_{n}(s,u_{n},u_{n+1})$ to be the fractional part in $(-1/2,1/2]$ of $\alpha (\langle s,h_{n} \rangle-\rho_{n+1}(u_{n+1})+\rho_{n}(u_{n}))$. Clearly 
$\vvert \theta_{n}(s,u_{n},u_{n+1}) \vvert=| \theta_{n}(s,u_{n},u_{n+1})|$.

We have that, 
\begin{align*}
|e^{2i\pi\theta_{n}(s,u_{n},u_{n+1})}-1|^{2} 
&=(e^{2i\pi\theta_{n}(s,u_{n},u_{n+1})}-1)(e^{-2i\pi\theta_{n}(s,u_{n},u_{n+1})}-1)\\
&=2 (1-\cos( \theta_{n}(s,u_{n},u_{n+1}))). 
\end{align*}
Consider a very small $\varepsilon >0 $. Then, by classical geometric arguments, there exists $0< \delta < \varepsilon$ such that for $n$ large enough and 
$s \in \sufijocbb{n}{n+1}{u_{n}}{u_{n+1}}$ with 
$\vvert  \theta_{n}(s,u_{n},u_{n+1}) \vvert \leq \delta$
we have that 
$$\left |\frac{\vvert  \theta_{n}(s,u_{n},u_{n+1}) \vvert^{2}}{2 (1-\cos( \theta_{n}(s,u_{n},u_{n+1})))}-1\right | \leq \varepsilon , $$
and thus 
\begin{align*}
(1-\varepsilon) |e^{2i\pi\theta_{n}(s,u_{n},u_{n+1})}-1|^{2}   
\leq \vvert  \theta_{n} (s,u_{n},u_{n+1}) \vvert^{2} 
\leq (1 + \varepsilon) |e^{2i\pi\theta_{n}(s,u_{n},u_{n+1})}-1|^{2} \ . 
\end{align*}

Also, if \eqref{eq:cnsvpmedible} holds, for any $n\geq 1$
$$
\frac{\# \{ s \in \sufijocbb{n}{n+1}{u_{n}}{u_{n+1}} \ ; \ \vvert  \theta_{n}(s,u_{n},u_{n+1}) \vvert > \delta \}}{M_{n+1}(u_n,u_{n+1})} 
\leq \frac{1}{f(\delta)} \cdot \varepsilon_n \ , 
$$
where $(\varepsilon_n ;  n \in \NN)$ is summable and $f:\RR \to \RR$ is a 
non-decreasing continuous function with $\lim_{t\to 0} f(t)=0$. 
Then, condition \eqref{eq:cnsvpmedible} implies, 
\begin{equation}
\label{eq:cnsvpmediblebis}
\sum_{n\geq 1} \frac{1}{M_{n+1}(u_{n},u_{n+1})} \sum_{s\in \sufijocbb{n}{n+1}{u_{n}}{u_{n+1}}} 
\vvert  \theta_{n}(s,u_{n},u_{n+1}) \vvert ^{2} < +\infty \ .
\end{equation}
In fact conditions \eqref{eq:cnsvpmedible} and 
\eqref{eq:cnsvpmediblebis} are equivalent (the proof is left to the reader). 
\medskip

In the linearly recurrent case, one easily gets that condition \eqref{eq:cnsvpmediblebis} is equivalent to $$\sum_{n\geq 1} \vvert \alpha h_{n} \vvert^{2} < +\infty \ ,$$
which is the necessary and sufficient condition for $\lambda$ to be an eigenvalue for any ergodic probability measure $\mu$ proved in \cite{necesariasuficiente}. 

%%%%%%%%%%%%%%%%%
\subsubsection{Markov process}
%%%%%%%%%%%%%%%%%
In \cite{necesariasuficiente} has been observed that $(\tau_n; n \in \NN)$ is a Markov chain with respect to any invariant measure of $(X,T)$.
It has been shown to play an important role in the study of eigenvalues (see also \cite{alistecoronel}).
This is mainly due to the mixing condition given below (Lemma \ref{lem:geometricdeacrieasing}).

First we set the following notations and assumptions. 

\begin{enumerate}
\item For $m,n \in \NN$ with $m<n$ define the following stochastic matrices: for $u,v \in I_\mu$ 
\begin{align*}
q_{m,n}(u,v)&=\mu(\tau_{n}=v|\tau_{m}=u) \\
&=\frac{\mu_{n}(v)}{\mu_{m}(u)} \matrizp{m}{n}{u}{v} \\
&= \frac{\mu(\tau_{n}=v)}{\mu(\tau_{m}=u)} \cdot \frac{h_{m}(u)}{h_{n}(v)} \cdot \matrizp{m}{n}{u}{v}. 
\end{align*}
\item 
Our diagram has been telescoped in such a way that for all $u,v\in I_{\mu}$ and $m,n \in \NN$ with $m<n$
$$q_{m,n}(u,v)\geq \delta_0^{2}/2, \quad \mu(\tau_{n}=v)\geq \delta_0 ,$$
where $\delta_0$ is the constant given by the cleanness of the diagram. (Observe this is always possible thanks to Lemma \ref{lemma:comportamiento_cuocientes}.) 
\item For $n >1$ define $\zeta(q_{n-1,n})=1-\min_{u,v\in I_{\mu}}(q_{n-1,n}(u,v))$. By  assumption (2) we have that 
$\zeta(q_{n-1,n})\leq 1- \delta_0^{2}/2 < 1$. Let $\zeta=\sup_{n\in \NN} \zeta(q_{n-1,n})$. 
\end{enumerate}

Using Lemma 5.3 in \cite{alistecoronel} we get,
\begin{lemma} For $m,n \in \NN$ with $m<n$
$$\max_{u,u',v \in I_{\mu}} \left | q_{m,n}(u,v)-q_{m,n}(u',v)\right | \leq \zeta^{n-m} \ .$$
\end{lemma}

From here we deduce, 

\begin{lemma} 
\label{lem:geometricdeacrieasing}
For $m,n \in \NN$ with $m<n$ and $u,v \in I_{\mu}$ we have 
$$\left | \mu(\tau_{n}=v | \tau_{m}=u)- \mu(\tau_{n}=v) \right | \leq  \zeta^{n-m}.$$
\end{lemma}
\begin{proof}
Recall we are assuming that the system is of exact finite rank $d$. 
Let $u,v \in I_{\mu}$. We have

\begin{align*}
& |\mu(\tau_{n}=v | \tau_{m}=u)- \mu(\tau_{n}=v)| \\
=&|\mu(\tau_{n}=v | \tau_{m}=u)- \sum_{u'\in I_{\mu}} \mu(\tau_{n}=v | \tau_{m}=u') \mu(\tau_{m}=u')| \\
=&|\sum_{u'\in I_{\mu}} \left ( \mu(\tau_{n}=v | \tau_{m}=u)-(\tau_{n}=v | \tau_{m}=u') \right ) \mu(\tau_{m}=u')|\\
\leq & \sum_{u'\in I_{\mu}} \zeta^{n-m} \mu(\tau_{m}=u')= \zeta^{n-m},
\end{align*}
where in the last equality we used that the system is of exact rank and thus
$\sum_{u'\in I_{\mu}} \mu(\tau_{m}=u')=1$.
\end{proof}
Since we are considering clean diagrams, in the non exact finite rank case the last equality can be easily replaced by an inequality with: $\# I_\mu \cdot \zeta^{n-m} + \varepsilon_{m}$, where $\varepsilon_{m}$ goes to $0$ uniformly in $m,n\in \NN$ with $m<n$. For the moment we do not know how to improve this bound in the non exact rank case. 

\subsubsection{Fundamental random variable $X_{n}$}
Recall we are assuming $I_{\mu}=\{1,\ldots,d\}$. To make levels explicit depending on the context we will write $V_n$ instead of $I_\mu$ or $\{1,\ldots,d\}$.  

Define for each $n\geq 1$ and $x\in X$: 
${\btheta}_{n}(x)=\theta_{n}(s_{n}(x),\tau_{n}(x),\tau_{n+1}(x))$. Consider the random variable
$X_{n}=\btheta_{n}-\EE_{\mu}(\btheta_{n})$ and decompose it as $X_{n}=Z_{n}+Y_{n}$ where 
$Z_{n}=\btheta_{n}-\EE_{\mu}(\btheta_{n}| {\mathcal T}_{n})$ and 
$Y_{n}=\EE_{\mu}(\btheta_{n}| {\mathcal T}_{n}) - \EE_{\mu}(\btheta_{n})=\EE_{\mu}(X_{n}|{\mathcal T}_{n})$. Recall that $\T_n$ is the sigma algebra generated by the Kakutani-Rohlin partition of level $n$. 
\medskip

We prove the convergence in $L^2 (X,\mu)$ of $\sum_{n\geq 1} Z_{n}$ and $\sum_{n\geq 1} Y_{n}$, and thus of $\sum_{n\geq 1} X_{n}$. 
\medskip

\paragraph{\it Some preliminary bounds:} notice that 
$$\btheta_{n}=\sum_{u\in V_{n}} \sum_{v\in V_{n+1}} \sum_{s\in \sufijocbb{n}{n+1}{u}{v}} 
\theta_{n}(s,u,v) \, \, {\bf 1}_{\{\tau_{n}=u,\tau_{n+1}=v,s_{n}=s\}}.$$ 
Then, 
\begin{align*}
&\EE_{\mu}(\btheta_{n}| {\mathcal T}_{n})\\
=&\sum_{u\in V_{n}} {\bf 1}_{\{\tau_{n}=u\}} 
\sum_{v\in V_{n+1}} \frac{h_{n}(u)\mu_{n+1}(v)}{\mu(\tau_{n}=u)}
\sum_{s\in \sufijocbb{n}{n+1}{u}{v}} \theta_{n}(s,u,v) \\
=&\sum_{u\in V_{n}} {\bf 1}_{\{\tau_{n}=u\}} 
\sum_{v\in V_{n+1}} \frac{\mu(\tau_{n+1}=v)}{\mu(\tau_{n}=u)} \frac{h_{n}(u)}{h_{n+1}(v)}\sum_{s\in \sufijocbb{n}{n+1}{u}{v}} \theta_{n}(s,u,v) \\
=&\sum_{u\in V_{n}} {\bf 1}_{\{\tau_{n}=u\}} 
\sum_{v\in V_{n+1}} \frac{\mu(\tau_{n+1}=v)}{\mu(\tau_{n}=u)} A_{n+1}(u,v) 
\frac{1}{M_{n+1}(u,v)}\sum_{s\in \sufijocbb{n}{n+1}{u}{v}} \theta_{n}(s,u,v), 
\end{align*}
where $A_{n+1}(u,v)=\frac{h_{n}(u)}{h_{n+1}(v)}\cdot M_{n+1}(u,v)$.
Thus, 
\begin{align*}
& \EE_{\mu}(\btheta_{n})= \EE_{\mu}(\EE_{\mu}(\btheta_{n}| {\mathcal T}_{n})) \\
=&\sum_{u\in V_{n}}  \mu(\tau_{n}=u) 
 \sum_{v\in V_{n+1}}  \frac{\mu(\tau_{n+1}=v)}{\mu(\tau_{n}=u)} A_{n+1}(u,v)  
 %\left (
  \frac{1}{M_{n+1}(u,v)}\sum_{s\in \sufijocbb{n}{n+1}{u}{v}} \theta_{n}(s,u,v)  
  %\right )
  .
\end{align*}
Similarly, 
\begin{align*}
& \EE_{\mu}(\btheta_{n}^{2}| {\mathcal T}_{n}) \\
= & \sum_{u\in V_{n}} {\bf 1}_{\{\tau_{n}=u\}} 
\sum_{v \in V_{n+1}} \frac{\mu(\tau_{n+1}=v)}{\mu(\tau_{n}=u)} A_{n+1}(u,v) 
\frac{1}{M_{n+1}(u,v)}\sum_{s\in \sufijocbb{n}{n+1}{u}{v}} \theta_{n}^{2}(s,u,v), 
\end{align*}
and 
\begin{align*}
& \EE_{\mu}(\btheta_{n}^{2})=\EE_{\mu}(\EE_{\mu}(\btheta_{n}^{2}| {\mathcal T}_{n}))\\
=&\sum_{u\in V_{n}}  \mu(\tau_{n}=u) 
\sum_{v\in V_{n+1}} \frac{\mu(\tau_{n+1}=v)}{\mu(\tau_{n}=u)} A_{n+1}(u,v)  \frac{1}{M_{n+1}(u,v)}\sum_{s\in \sufijocbb{n}{n+1}{u}{v}} \theta_{n}^{2}(s,u,v) .
\end{align*}

Therefore, 
\begin{align*}
|Y_{n}|=& |\EE_{\mu}(X_{n}| {\mathcal T}_{n})| \\
\leq & \sum_{u\in V_{n}} |{\bf 1}_{\{\tau_{n}=u\}}-\mu(\tau_{n}=u)|  \sum_{v\in V_{n+1}} \frac{\mu(\tau_{n+1}=v)}{\mu(\tau_{n}=u)}  A_{n+1}(u,v) \\
& \times \left |   \frac{1}{M_{n+1}(u,v)}\sum_{s\in \sufijocbb{n}{n+1}{u}{v}} \theta_{n}(s,u,v)  \right | \\
\leq &  \sum_{u\in V_{n}} 2 \sum_{v\in V_{n+1}}  \frac{1}{\delta_0} \cdot 1 \cdot \frac{1}{M_{n+1}(u,v)}\sum_{s\in \sufijocbb{n}{n+1}{u}{v}} |\theta_{n}(s,u,v) | \ , \\
\end{align*}
where $\delta_0$ is the constant defining the cleanness property. 
But, from hypothesis \eqref{eq:cnsvpmediblebis}, 
the term in the second sum is bounded by some $\xi_{n}$ where $\sum_{n \geq 1} \xi_{n}^{2} <+\infty$. We conclude that $|Y_{n}| \leq K\xi_{n},$
where $K=2d^{2}/\delta_0$.

\medskip

Below we will manipulate a lot of constants depending neither on variables nor on indices. We will call them universal constants and denote them by $K$. 
\medskip


For $Z_n$ a classical computation gives, 
\begin{align*}
\EE_{\mu}(Z^{2}_{n})&= \EE_{\mu}((\btheta_{n}-\EE_{\mu}(\btheta_{n}|{\mathcal T}_{n}))^{2}) =\EE_{\mu}(\btheta_{n}^{2})- \EE_{\mu}(\btheta_{n} \EE_{\mu}(\btheta_{n}|{\mathcal T}_{n})).
\end{align*}
Thus, using a similar argument as in the bound for $|Y_{n}|$, we get
\begin{align*}
\EE_{\mu}(Z^{2}_{n})&\leq \EE_{\mu}(\btheta_{n}^{2}) + |\EE_{\mu}(\btheta_{n} \EE_{\mu}(\btheta_{n}|{\mathcal T}_{n}))| \leq \EE_{\mu}(\btheta_{n}^{2}) + \EE_{\mu}(|\btheta_{n}| |\EE_{\mu}(\btheta_{n}|{\mathcal T}_{n})|) \\
& \leq K_{1} \xi_{n}^{2} + \EE_{\mu}(|\btheta_{n}|) K_{2} \xi_{n} 
\leq K_{1} \xi_{n}^{2} + K_{3}\xi_{n} K_{2} \xi_{n}= K_4 \xi_{n}^2, \\
\end{align*}
for some universal constants $K_{1}, K_{2}, K_{3}, K_4$. Therefore,
$$
\sum_{n\geq 1} \EE_{\mu}(Z^{2}_{n}) \leq K \sum_{n\geq 1} \xi_{n}^{2} < +\infty.
$$ 

\paragraph{\it The series $\sum_{n\geq 1} Z_n$ converges in $L^2 (X,\mu)$}
Let $m,n \in \NN$ with $m<n$. We have that 
$\EE_{\mu}(Z_{n}|{\mathcal T}_{n})=\EE_{\mu}(\btheta_{n}-\EE_{\mu}(\btheta_{n}|{\mathcal T}_{n})|{\mathcal T}_{n})=0$. Then, since ${\mathcal T}_{n}$ is finer than ${\mathcal T}_{m}$, we get
$$\EE_{\mu}(Z_{m}Z_{n})=\EE_{\mu}(\EE_{\mu}(Z_{m}Z_{n}|{\mathcal T}_{n}))=\EE_{\mu}(Z_{m}\EE_{\mu}(Z_{n}|{\mathcal T}_{n}))=0.$$
 
This implies that 
$$\langle \sum_{i=m}^{n} Z_{i}, \sum_{j=m}^{n} Z_{j} \rangle = \sum_{i=m}^{n} \sum_{j=m}^{n} \langle Z_{i}, Z_{j} \rangle = \sum_{\ell=m}^{n} ||Z_{\ell}||^{2}_{L^2 (X,\mu)} \leq K \sum_{\ell=m}^{n} \xi^{2}_{\ell},$$
and proves that $(\sum_{n\geq 1} Z_{n})$ converges in $L^2 (X,\mu)$.
\medskip 

\paragraph{\it The series $\sum_{n\geq 1} Y_n$ converges in $L^2 (X,\mu)$} 
We follow the scheme given in \cite{necesariasuficiente}. 
We have that $Y_{n}=\EE_{\mu}(X_{n}|{\mathcal T}_{n})=\sum_{v \in V_{n}} {\bf 1}_{\{\tau_{n}=v\}} \, y_{n}(v)$, where $y_{n}(v)$ are constant values. 
Let $0\leq k \leq n$. Then, 
\begin{align*}
\EE_{\mu}(Y_{n}|{\mathcal T}_{n-k})
&=\EE_{\mu}\left(\sum_{v \in V_{n}} {\bf 1}_{\{\tau_{n}=v\}} \, y_n{(v)}\mid{\mathcal T}_{n-k}\right)
=\sum_{v\in V_{n}}\EE_{\mu}({\bf 1}_{\{\tau_{n}=v\}}|{\mathcal T}_{n-k})  \, y_{n}(v)\\
&=\sum_{v\in V_{n}}\sum_{u\in V_{n-k}} {\bf 1}_{\{\tau_{n-k}=u\}} \frac{\int_{\{\tau_{n-k}=u\}} {\bf 1}_{\{\tau_{n}=v \}} d\mu}{\mu(\tau_{n-k}=u)} \, y_{n}(v)\\
&=\sum_{u\in V_{n-k}} {\bf 1}_{\{\tau_{n-k}=u\}} \sum_{v\in V_{n}} \mu(\tau_{n}=v|\tau_{n-k}=u) y_{n}(v) . \\
\end{align*}

We have that $\EE_{\mu}(Y_{n})=\EE_{\mu}(X_{n})=\EE_{\mu}(\btheta_{n}-\EE_{\mu}(\btheta_{n}))=0$. So, 
$\sum_{v\in V_{n}} \mu(\tau_{n}=v) y_{n}(v)=0$ and thus
$$\EE_{\mu}(Y_{n}|{\mathcal T}_{n-k})=\sum_{u\in V_{n-k}} {\bf 1}_{\{\tau_{n-k}=u\}} 
 \sum_{v\in V_{n}} (\mu(\tau_{n}=v|\tau_{n-k}=u) -\mu(\tau_{n}=v)) y_{n}(v)  .$$
From this expression and Lemma \ref{lem:geometricdeacrieasing} we get
\begin{align*}
|\EE_{\mu}(Y_{n}|{\mathcal T}_{n-k})| 
&\leq  \sum_{u\in V_{n-k}} \sum_{v\in V_{n}} |\mu(\tau_{n}=v|\tau_{n-k}=u) -\mu(\tau_{n}=v)| |y_{n}(v)|  
\leq  K' \zeta^{k} \xi_{n},
\end{align*} 
where $K$ is a universal constant and we have used that $|Y_{n}|\leq K \xi_{n}$.
From here we deduce that 
\begin{align*}
|\EE_{\mu}(Y_{n} Y_{n-k})| = &
|\EE_{\mu}(\EE_{\mu}(Y_{n} Y_{n-k}|{\mathcal T}_{n-k}))| 
=|\EE_{\mu}( Y_{n-k} \EE_{\mu}(Y_{n} |{\mathcal T}_{n-k}))| \\
\leq & K \zeta^{k} \xi_{n} K \xi_{n-k}=K \zeta^{k} \xi_{n} \xi_{n-k}.
\end{align*}

From previous discussions and formulas we get,
\begin{align*}
\langle \sum_{i=m}^{n} Y_{i}, \sum_{j=m}^{n} Y_{j}\rangle 
&= \sum_{i=m}^{n} \sum_{j=m}^{n} \langle Y_{i}, Y_{j} \rangle 
\leq K \sum_{i=m}^{n} \sum_{j=m}^{n} \zeta^{|i-j|} \xi_{i} \xi_{j} 
\leq K \sum_{k=0}^{n-m} \zeta^{k} \sum_{l=m}^{n-k} \xi_{l} \xi_{l+k} \\
&\leq K \sum_{k=0}^{n-m} \zeta^{k} \sum_{l=m}^{n} \xi_{l}^{2} 
\leq K \frac{\zeta^{n-m+1}-1}{\zeta-1} \sum_{l=m}^{n} \xi_{l}^{2} . \\
\end{align*}
This implies that $\EE_{\mu}((\sum_{l=m}^{n}Y_{l})^{2})\leq K \sum_{l=m}^{n} \xi_{l}^{2}$. So 
$(\sum_{n\geq 1} Y_{n})$ converges in  $L^2 (X,\mu)$.
\medskip 

Finally, the conclusion from the last two computations is that  $\sum_{n\geq 1} X_{n}$ converges in  $L^2 (X,\mu)$ too.

\subsubsection{End of the proof of Theorem \ref{theo:cnsvpmedible}: construction of an eigenfunction for $\lambda$} From previous discussion we have that 
$f_n=\exp(2i\pi  \sum_{k=1}^{n-1} X_k)$ converges in $L^2 (X,\mu)$ to some function $f$.
For $n\geq 1$ and $x \in X$ we have 
\begin{align*}
&f_n(Tx)/f_n(x) \\ 
=&\exp 
\left(
2i\pi  
\sum_{k=1}^{n-1} \theta_k(s_k(Tx),\tau_k(Tx),\tau_{k+1}(Tx)) -\theta_k(s_k(x),\tau_k(x),\tau_{k+1}(x))  
\right) \\
=&\exp 
\left(
2i\pi \alpha \sum_{k=1}^{n-1} (\langle s_k(Tx),h_k \rangle -  \langle s_k(x),h_k \rangle  
\right.\\ 
& \left.
- 2i\pi \alpha \sum_{k=1}^{n-1} (\rho_{k+1}(\tau_{k+1}(Tx))-\rho_{k+1}(\tau_{k+1}(x))-\rho_k(\tau_k(Tx))+\rho_k(\tau_k(x)) ) 
\right) \\
= &\exp 
( 
2i\pi \alpha \sum_{k=1}^{n-1} (\langle s_k(Tx),h_k \rangle - 
\langle s_k(x),h_k \rangle  \\
& - 2i\pi \alpha ( \rho_{n}(\tau_{n}(Tx))-\rho_{n}(\tau_{n}(x))-\rho_1(\tau_1(Tx))+\rho_1(\tau_1(x))) ) \\
=&\exp (2i\pi \alpha ( r_n(Tx)-r_n(x) -\rho_{n}(\tau_{n}(Tx))+\rho_{n}(\tau_{n}(x)))), 
\end{align*}
where in the last equality we have assumed without loss of generality that $\rho_1$ is a constant function. 

Now, if $x$ is not in the base $\cup_{v\in V_n} B_n(v)$ of level $n$, then 
$r_n(Tx)-r_n(x)=-1$ and $\tau_{n}(Tx)=\tau_{n}(x)$. 
Thus, in this case, $f_n(Tx)/f_n(x)=\lambda^{-1}$. 
Since $\lim_{n\to +\infty}$ $\mu(\cup_{v\in V_n} B_n(v))=0$ we conclude that 
$f\circ T=\lambda^{-1} f$ in $L^2 (X,\mu)$. This proves that condition \eqref{eq:cnsvpmedible} is a sufficient condition for $\lambda$ to be an eigenvalue for $\mu$.
$\square$



 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Examples} 
\label{examples}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we illustrate with simple examples the use of the main theorems of this article. In all the examples eigenvalues are known, the purpose is to show how the combinatorial objects required by the theorems are controlled. 

%%%%%%%%%%%%%%%%%%%
\subsection{Sturmian systems}
%%%%%%%%%%%%%%%%%%%
Let $(\RR/\ZZ, S_{\alpha})$ be an irrational rotation of the circle, where $S_{\alpha}x=x+\alpha$ (mod $\ZZ$) with $\alpha$ an irrational number. This system is minimal and uniquely ergodic, being the Lebesgue measure on $[0,1)$ the unique invariant measure. Also, $\lambda=\exp(2i \pi \beta)$ is a (continuous) eigenvalue of the system if and only if $\beta\in\set{m\alpha + n\talque m,n\in\ZZ}$. Let $(X,T)$ be the \emph{Sturmian system} associated to $(\RR/\ZZ, S_{\alpha})$. That is, the subshift generated by the natural coding of the orbits of the irrational rotation on the partition $\{[0,1-\alpha),  [1-\alpha, 1)\}$. This system is an asymptotic extension of the irrational rotation and has the same continuous eigenvalues. 

We provide a Bratteli-Vershik representation of $(X, T)$ in order to show how Theorem \ref{theo:nec-suff-cont} generates a natural procedure to recover the continuous eigenvalues of this system. 

Let us recall some basic facts about simple continued fractions (see \cite[Ch. X]{hardywright} for details). Write $\alpha=[0;a_1,a_2,\ldots]$ in terms of its simple continued fraction expansion, where $(a_n\talque n\geq 1)$ is a sequence of positive integers. Also, define the increasing sequences of positive integers $(p_n\talque n\geq 0)$ and $(q_n\talque n\geq 0)$ recursively by 
\begin{equation}\label{eq:recursion_aproximantes}
\begin{array}{llll}
p_0=0, & p_1=1, & p_n=a_np_{n-1} + p_{n-2} & {\rm for} \; n>1, \\
q_0=1, & q_1=a_1, & q_n=a_nq_{n-1} + q_{n-2} & {\rm for} \; n>1.
\end{array}
\end{equation}
Then,

\begin{equation*}
  \alpha = \cfrac{1}{a_1
          + \cfrac{1}{a_2 + \cdots\; } } \ , \ \frac{p_n}{q_n} = \cfrac{1}{a_1
             + \cfrac{1}{\cdots + \displaystyle\frac{1}{a_n}} } \ , \end{equation*}
for $n>0$, and this defines a rational sequence converging to $\alpha$ which rate of convergence is given by
\begin{equation}\label{eq:distancia_aproximantes}
\left|\alpha - \frac{p_n}{q_n}\right|<\frac{1}{q_{n}^{2}}.
\end{equation}
Also, the subsequences $(p_{2n}/q_{2n}\talque n\geq 0)$ and $(p_{2n+1}/q_{2n+1}\talque n\geq 0)$ are monotonically increasing and decreasing respectively.
Finally, if $(\phi_n\talque n\geq 0)$ stands for the Fibonacci sequence, then $q_n\geq \phi_n$ for all $n\geq 0$, and therefore
$\sum_{n\geq 0}\frac{1}{q_n} <+\infty.$


Using a slight modification of Rauzy-Veech induction (see for example \cite{viana_iet_ergodic}) we get a Bratteli-Vershik representation $B=(V,E,\preceq)$ of $(X,T)$ such that: $\#V_n = 2$ for all $n\geq 1$; matrices are given by $M_1=h_1=(1,1)$, $M_2=\left(\begin{smallmatrix}
a_1&a_1 - 1\\ 1&1
\end{smallmatrix}\right)$ and $M_{n+1}=\left(\begin{smallmatrix}
1&1\\ a_n&a_n-1
\end{smallmatrix}\right)$ for all $n\geq 2$, and the local orders are
as in the next figure.
\begin{figure}[ht]
\begin{center}
\input{fig_ordenes_rotacion.tex}
\label{fig:ordenes_rotacion}
\caption{Levels of the Bratteli-Vershik representation $B$ , showing their respective orders.}
\end{center}
\end{figure}
\medskip

Notice that this representation could not be simple in some cases (when $a_n=1$ for one or several values of $n$). Nevertheless, a telescoping of this diagram taking specific gaps of length not bigger than 2 turns out to be a simple representation. 

As discussed before, in the case of continuous eigenvalues, to find candidates we use conditions \eqref{eq:eta} and \eqref{eq:eta1}. 

\begin{prop} Let $\lambda=\exp(2i \pi \beta)$ be a continuous eigenvalue of $(X,T)$. Then $\beta\in\set{m\alpha + n \talque m,n \in\ZZ}$. 
\end{prop}
\begin{proof}
From conditions \eqref{eq:eta} and \eqref{eq:eta1}, and the fact that incidence matrices of the system have determinant equal to $\pm 1$, one gets row vectors $v\in \RR^{2}$ and $w\in \ZZ^{2}$ such that
$
\beta h_1  = v + w \; \; \; {\rm and} \; \; \; vP_{n}\xrightarrow[n\to+\infty]{} 0. 
$
Also, from \eqref{eq:eta2} we get that $v$ is orthogonal to $\mu_1$. 
By construction $\mu_1=\left(\begin{smallmatrix}
1-\alpha \\ \alpha \end{smallmatrix}\right)$ (see Section \ref{subsubsec:invmeas} for the definition of $\mu_1$), so it exists $\gamma\in\RR$ such that $v=\gamma\left(\begin{smallmatrix}
\alpha \\ \alpha-1
\end{smallmatrix}\right)$. Then,
$\beta h_1 = v + w
= \gamma\left(\begin{smallmatrix}
\alpha \\ \alpha-1
\end{smallmatrix}\right) + w,$
and from this we have that $\gamma\in\ZZ$ and $\beta$ has the desired form. 
\end{proof}
\medskip

Now we have the candidates it only remains to check that $\exp(2 i \pi\alpha)$ is a continuous eigenvalue of $(X,T)$. We can work with the (possibly non-simple) Bratteli-Vershik representation $B$. According to Remark \ref{rmrk:cond_suf_sin_simple} it is enough to see whether $\alpha$ satisfies condition (2) in Theorem  \ref{theo:nec-suff-cont} for this representation.



\begin{prop} With the definition above, $\lambda=\exp(2i \pi \alpha)$ is a continuous eigenvalue of the minimal Cantor system $(X,T)$.
\end{prop}
\begin{proof}
Using \eqref{eq:recursion_aproximantes} it is easy to check by induction that $h_n=(q_{n-1} + q_{n-2}, q_{n-1})$ for $n>1$.
We are going to use condition (2) of Theorem \ref{theo:nec-suff-cont}, so it is enough to prove  
\begin{equation}\label{eq:cota_cada_sumando_q_n-2}
{\rm for}\; \;  n>4 \; \; {\rm and} \; \; x\in X, \; \vvert \la s_n(x),\alpha h_n\ra \vvert < \frac{1}{q_{n-2}}.
\end{equation}
Indeed, let us take $x\in X$ and $n>4$. 
Notice that we have $1/q_{n-2}\leq 1/2$.

Now, by the order associated to the Bratteli diagram we have that $s_n(x)=(0,0)$ or $(0,1)$, or it belongs to $\set{(1,i)\talque 0\leq i\leq a_{n-1}-1}$. We are going to calculate separately in each case.

For $s_n(x)=(0,0)$, $\vvert \la s_n(x),\alpha h_n\ra \vvert = 0$ and \eqref{eq:cota_cada_sumando_q_n-2} follows trivially.
For $s_n(x)=(0,1)$, $\la s_n(x),\alpha h_n\ra = \alpha q_{n-1}$. From \eqref{eq:distancia_aproximantes} we obtain $|\alpha q_{n-1} - p_{n-1}| < 1/q_{n-1}$, and \eqref{eq:cota_cada_sumando_q_n-2} follows because $(q_{n}\talque n\geq 0)$ is strictly increasing.
For $s_n(x)=(1,i)$ with $0\leq i\leq a_{n-1}-1$, 
$
\la s_n(x),\alpha h_n\ra = \alpha q_{n-2} + \alpha(i+1)q_{n-1}.
$ 
Let us assume first that $n$ is even. From the different behavior of the odd and even subsequences of $(p_k/q_k\talque k\geq 0)$ and \eqref{eq:distancia_aproximantes} we have
\begin{align*}
0&< \alpha q_{n-2} - p_{n-2} < \frac{1}{q_{n-2}} \; \; {\rm and}\\
0&< (i+1)p_{n-1} - \alpha(i+1) q_{n-1} < \frac{i+1}{q_{n-1}} \leq \frac{a_{n-1}}{q_{n-1}} < \frac{1}{q_{n-2}},
\end{align*}
where the very last inequality comes from the recursive equation \eqref{eq:recursion_aproximantes}. Then
$$
\vvert \alpha q_{n-2} + \alpha(i+1)q_{n-1} \vvert <\frac{1}{q_{n-2}},
$$
and \eqref{eq:cota_cada_sumando_q_n-2} follows (the analysis for $n$ odd is analogous).
\end{proof}

%%%%%%%%%%%%%%%%%%%
\subsection{Eigenvalues of Toeplitz systems}\label{sub sub:toeplitz}
%%%%%%%%%%%%%%%%%%%

A Bratteli-Vershik system is called of \emph{Toeplitz-type} if its corresponding Bratteli diagram has the \emph{equal path number} property. That is, the number of edges ending at some fixed vertex is constant within the respective level: for any $n\geq 1$, $\#\set{e\in E_n\talque \range(e)=u}$ is a positive integer independent of $u\in V_{n}$. This integer will be denoted by $q_n$ and $(q_n\talque n\geq 1)$ is called \emph{the characteristic sequence} of the system. We also define the quantities $p_n=q_1q_2\cdots q_n$ for $n\geq 1$ and $q_{m,n}=q_{m+1}q_{m+2}\cdots q_{n}$ for $n>m\geq 1$. We notice that for $n>m\geq 1$, 
$$
h_m(u)/h_n(v)=1/q_{m,n} \; \text{  for all  } u\in V_m \text{  and  } v\in V_n.
$$

It is an easy exercise to show that odometers (\emph{i.e}, equicontinuous minimal Cantor systems) have a Bratteli-Vershik representation of Toeplitz-type.
In \cite{gjtoeplitz} it is shown that every Toeplitz subshift can be represented by an expansive Bratteli-Vershik system of Toeplitz-type. 
On the other side, from \cite{DM} we know that non-equicontinuous finite rank Bratteli-Vershik systems of Toeplitz-type are Toeplitz subshifts. 

Let $(X,T)$ be a Toeplitz system of finite rank $d$ given by a proper Toeplitz-type Bratteli-Vershik representation. As is customary (see \cite{rangofinito} and \cite{dfm}), in the case of Toeplitz systems of finite rank we let $\overline{s}_{m,n}(x)$ stand for $\la s_{m,n}(x), h_1\ra$ for $x\in X$ and $n>m\geq 1$. We have $\la s_{m,n}(x), h_n\ra= p_n\overline{s}_{m,n}(x)$, and that for each $n\geq 1$ the function $\overline{s}_{n}:=\overline{s}_{n,n+1}$ takes all the values between $0$ and $q_{n+1}-1$. Also we define the set $\overline{S}_{m,n}(u,v)=\set{\overline{s}_{m,n}(x)\talque \tau_m(x)=u, \tau_{n}(x)=v}$.

%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Continuous eigenvalues}
%%%%%%%%%%%%%%%%%%%%%
The following characterization for continuous eigenvalues of a Toeplitz system is well-known (see \cite{Keane,Williams} or Theorem 25 in \cite{rangofinito}). We provide a proof here to emphasize the use of Theorem \ref{theo:nec-suff-cont}. 
We assume that the Bratteli-Vershik representation of Toeplitz-type has characteristic sequence $(q_n\talque n\geq 1)$. 

\begin{theo}
The complex number $\exp(2 i \pi \alpha)$ is a continuous eigenvalue of the Toeplitz system $(X,T)$  if and only if $\alpha = a/p_N$, for some $a\in\ZZ$ and $N\geq 1$.
\end{theo}
\begin{proof}
Let $\exp(2i \pi \alpha)$ be a continuous eigenvalue of the system with $\alpha\in [0,1)$. Then, by \eqref{eq:eta}, for any $n\geq 1$, $\alpha h_n= \alpha p_n (1,\ldots,1)=\eta_n + \nu_n$
with $\eta_n \to 0$ as $n\to + \infty$ and $\nu_n$ an integer vector. As remarked before, from \eqref{eq:eta1} we deduce that $ \eta_n \cdot \mu_n =0$ for any large $n \in \NN$. 

Now, since for all large $n \in \NN$ we have that $h_n=p_n (1,\ldots,1)$, then $\eta_n= \bar\eta_n(1,\ldots,1)$ and $\nu_n=\bar\nu_n (1,\ldots,1)$ for some real numbers $\bar\eta_n$ and $\bar\nu_n$. Hence, 
$ \eta_n \cdot \mu_n =\bar\eta_n  (1,\ldots,1) \cdot \mu_n =0$. This implies that $\bar\eta_n=0$ and thus $\alpha= \bar\nu_n/p_n$ for all large enough $n\in \NN$. So $\alpha$ has the desired form.  

On the other hand, if $\alpha=a/p_N$ for some $a\in\ZZ$ and $N\geq 1$, then $\vvert \alpha p_n\overline{s}_n(x)\vvert=0$ for all $n>N$ and $x\in X$, and therefore condition (2) of Theorem \ref{theo:nec-suff-cont} holds. 
\end{proof}

By Corollary \ref{coro:rootsofunity}, all roots of the unity that satisfy \eqref{eq:eta} will be automatically continuous eigenvalues of the system. So the proof of the last part is not necessary. We gave it by completeness.   

%%%%%%%%%%%%%%%%%%%
\subsubsection{Non-continuous eigenvalues of Toeplitz systems}
%%%%%%%%%%%%%%%%%%%
Let $(X,T)$ be a Toeplitz system of finite rank $d$ given by a proper Bratteli-Vershik representation of Toeplitz type. We continue with the same notations as above. 

From \cite[\S 7]{rangofinito} we know that an eigenvalue $\lambda=\exp(2i\pi \alpha)$ of  $(X,T)$ is necessarily  rational, {\it i.e.}, $\alpha=a/b$ with $(a,b)=1$, and $b$ should satisfy the inequality $b/(b,p_n)\leq d$ for all $n$ large enough, where $d$ is the rank of the system and $(p_n\talque n\geq 1)$ comes from the characteristic sequence of the system. Moreover, if $\lambda$ is a non-continuous eigenvalue associated to any ergodic measure, then $b/(b,p_n) > 1$.

Let us consider an ergodic measure $\mu$ of $(X,T)$. With the notation associated to Toeplitz systems of finite rank, condition (1) of Theorem \ref{theo:cns_sinrho} may be written as
\begin{equation}\label{eq:cns_sinrho_toeplitz}
\sum_{u\in I_{\mu}}\frac{1}{q_{m,n}}\Bigg|\sum_{s\in\overline{S}_{m,n}(u,v)}\!\!\!\lambda^{p_ms}\Bigg|\xrightarrow[m\to+\infty]{} 1, \; \text{for all } v\in I_{\mu},
\end{equation}
uniformly for $n>m$, and we have recovered the characterization given in \cite[Corollary 5]{dfm}.

It is interesting to notice that condition \eqref{eq:cns_sinrho_toeplitz} actually implies directly that $\lambda$ satisfies the inequality $b/(b,p_n)\leq d$ for all large enough $n$'s (it can be seen from \cite[Lemma 13(1)]{dfm}, taking a smaller $\varepsilon$ in its proof). So we can state the following version of \cite[Theorem 3]{dfm}, that characterizes non-continuous eigenvalues of Toeplitz systems of finite rank.
\begin{theo}\label{theo:cns_toeplitz_new}
The complex number $\lambda=\exp(2i\pi \alpha)$ is a non-continuous eigenvalue of $(X,T)$ with respect to an ergodic measure $\mu$ if and only if
\begin{itemize}
\item[(1)] $\alpha = a/b$, with $(a,b)=1$ and $b/(b,p_n)>1$ for all large enough $n$'s and
\item[(2)] for all $v\in I_{\mu}$,
$$
\sum_{u\in I_{\mu}}\frac{1}{q_{m,n}}\Bigg|\sum_{s\in\overline{S}_{m,n}(u,v)}\!\!\!\lambda^{p_ms}\Bigg|\xrightarrow[m\to+\infty]{} 1,
$$
uniformly for $n>m$.
\end{itemize} 
\end{theo}

{\small
\bibliography{biblio}{}
\bibliographystyle{amsalpha}
}

\end{document}

VP-31Mar2015.bbl
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\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
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