Meena Mani

Sebastian Kurtek

Anuj Srivastava

Christian Barillot

A COMPREHENSIVE RIEMANNIAN FRAMEWORK FOR THE ANALYSIS OF WHITE MATTER FIBER TRACTS

A quantitative analysis of white matter fibers is based on different physical features (shape, scale, orientation and position) of the fibers, depending on the specific application. Due to the different properties of these features, one usually designs different metrics and spaces to treat them individually. We propose a comprehensive Riemannian framework that allows a joint analysis of these features in a consistent manner. For each combination, we provide a formula for the distance, i.e. quantification of differences between fibers and a formula for geodesics, i.e. optimal deformations of fibers into each other. We illustrate this framework in the context of clustering fiber tracts from the corpus callosum and study the results from different combinations of features.

INTRODUCTION

The analysis of white matter fibers, of interest in applications as varied as the differential diagnosis of white matter diseases, neurosurgery or a study of brain connectvity, requires different techniques and approaches. Some of the more recent Diffusion Tensor MRI (DT-MRI) literature has addressed issues such as the clustering of fibers into anatomically meaningful bundles [START_REF] Zhang | Identifying white-matter fiber bundles in dti data using an automated proximity-based fiber-clustering method[END_REF], atlas building for group studies [START_REF] O'donnell | Automatic Tractography Segmentation Using a High-Dimensional White Matter Atlas[END_REF] and statistical methods for quantitative analysis [START_REF] Maddah | A Unified Framework for Clustering and Quantitative Analysis of White Matter Fiber Tracts[END_REF] or to evaluate fractional anisotropy, mean diffusivity and other measures of white matter integrity along tracts [START_REF] Goodlett | Group analysis of dti fiber tract statistics with application to neurodevelopment[END_REF].

Geometrically, white matter fibers reconstructed using DT-MRI, can be described as 3-dimensional open, continuous curves. These fibers, when viewed as curves, have certain physical features associated with them, namely shape, scale, orientation, and position. There have been several recent papers on shape analysis of continuous curves using a Riemannian framework [START_REF] Younes | A metric on shape space with explicit geodesics[END_REF]. This type of framework has many advantages: (1) It provides techniques for comparing, matching, and deforming shapes of curves under the chosen metric. The correspondences for these tasks are established automatically. [START_REF] O'donnell | Automatic Tractography Segmentation Using a High-Dimensional White Matter Atlas[END_REF] It also provides tools for defining and computing statistical summaries of sample shapes for different shape Meena Mani was supported in part by an Université Européenne de Bretagne (UEB) grant.

classes [START_REF] Srivastava | Statistical shape analysis: Clustering, learning and testing[END_REF]. In the current problem, there is interest in including the other features -scale, orientation, and position -in the analysis, since this information can significantly aid the clustering, classifying, labeling and quantitative analysis of fiber tracts. It is therefore desirable to have a comprehensive Riemannian framework that can compare and quantify differences between these multiple features in a coherent way. In this conceptual paper, we define a sequence of feature spaces, each associated with a Riemannian metric, such that any desired combination of features of fibers can be analyzed using geodesics and geodesic distances. An important aspect of this work is that despite dealing with parameterized curves, the resulting geodesics and geodesic distances are invariant to the actual parametrization of the curves.

The rest of this paper is organized as follows. The mathematical framework for comparing curves using different features is laid out in Section 2. Experimental results demonstrating feasibility are presented in Section 3, and some concluding remarks are presented in Section 4.

MATHEMATICAL FRAMEWORK

Let β : [0, 2π] → R 3 be a parameterized curve, representing a fiber tract. We are interested in developing a Riemannian framework where we can study different physical features associated with β. An important requirement in this analysis is to obtain results that do not depend on any particular parameterization of β. If we use the standard L 2 metric for compar-

ing curves ( β 1 -β 2 2 = 2π 0 β 1 (t) -β 2 (t) 2 dt
), as is often done in quantitative analysis of fibers, then the results do not have the independence to the re-parameterizations. This forces us to choose novel mathematical representations of curves that allows us to have the desired invariance. In the following, we are going to present a sequence of representations (spaces and metrics) that capture different combinations of features and provide techniques for comparing curves according to those features. In particular, we are going to provide two things in each case: (i) a geodesic distance between curves that depends only on selected features (and is independent of the parameterization of curves), and (ii) a geodesic path between the two curves. The geodesic paths are useful for many reasons. Firstly, it provides a way of deforming one curve into the other in an optimal way. Secondly, it is fundamental to computing sample statistics, such as means and covariances, of curves.

Comparison using the complete feature set

We start by considering a situation where we are interested in comparing curves using all the four physical features -shape, scale, position and orientation. Define a representation of a curve β using a square-root function (SRF):

h(t) = β(t) β(t) , h : [0, 2π] → R 3 .
In order to compare any two curves, we will compare their SRFs. The metric that we use for this comparison is the L 2 metric. One advantage of this representation is its invariance to re-parameterization of curves. We explain this point further. Let γ : [0, 2π] → [0, 2π] be a smooth, one-to-one, and onto function; γ serves as a re-parameterization function for any curve. (Let Γ denote the set of all such functions.) For a curve β, the new curve β(t) ≡ β(γ(t)) is simply the old curve with a new parameterization. Also, for the re-parameterized curve β, the SRF is given by h(t) = γ(t)h(γ(t)). We often use (h, γ) to denote this re-parameterized SRF. Now, it can be shown that for any two curves β 1 , β 2 , with the corresponding SRFs h 1 and h 2 , and any γ ∈ Γ, we have that

(h 1 , γ) -(h 2 , γ) 2 = h 1 -h 2 2 .
Because of this equality, we can define a distance between the two curves as:

d a (β 1 , β 2 ) = min γ∈Γ ( h 1 -(h 2 , γ) 2 ) . ( 1
)
This minimization is performed using the standard dynamic programming (DP) algorithm, and it results in a quantification of differences in curves that is associated with all four features -shape, position, orientation, and scale. In addition to a proper distance, this framework also provides a geodesic path between the curves. In this case, the geodesic path is given by:

ψ(τ ) = (1 -τ )h 1 + τ (h 2 , γ * ) ,
where γ * is the optimal re-parameterization obtain earlier in minimization using DP.

Comparison using shape, scale and orientation

Suppose we want to compare curves using all features except their positions. Now we consider the task of comparing curves using three features (shape, orientation, and scale) and thus not including the position in the analysis. In this case we define a new representation of β, using the velocity function β, as follows: Given β : [0, 1] → R 3 , define

q(t) = β(t) || β(t)|| , q : [0, 2π] → R 3 .
This function is different from the earlier square-root function in that this definition is completely based on the velocity function β. Therefore, this function is called the square-root velocity function (SRVF) [START_REF] Joshi | A novel representation for Riemannian analysis of elastic curves in R n[END_REF]. Since this function is invariant to a global translation of β, any analysis based on this function will not dependent on the global coordinates of the curves. Note that the norm ||q(t)|| defines the square root of the instantaneous speed along the curve β. It is important to know that one can recover the original curve β, within a translation, using β(t) = t 0 ||q(s)||q(s)ds. If β = β • γ is a re-parameterized curve, then its SRVF is given by (q, γ) ≡ γ(t)q(γ(t)), where q is the SRVF of the original curve.

As earlier, it can be shown that for any curves β 1 , β 2 , with the corresponding SRVFs q 1 and q 2 , and any γ ∈ Γ, we have that (q 1 , γ) -(q 2 , γ) 2 = q 1q 2 2 . Once again, we can define a distance between the two curves as:

d b (β 1 , β 2 ) = min γ∈Γ q 1 -(q 2 , γ) 2 .
(

) 2
This minimization is performed using the DP algorithm, and it results in a quantification in differences in curves according to the remaining three features -shape, orientation, and scale. In this case, the geodesic path between the two curves is given by:

ψ(τ ) = (1 -τ )q 1 + τ (q 2 , γ * ) ,
where γ * is the optimal re-parameterization obtain the earlier minimization.

Comparison using shape and orientation

Now we consider a situation where the scales of observed curves are not important and we want to remove them from the analysis. In order to remove the influence of scales of curves in the quantitative analysis, we can rescale them to be of the same length, say 1. The mathematical representation and the Riemannian metric remains same as earlier except the set of SRVFs is reduced as a consequence of this rescaling. If the curve β is of length 1, then

1 0 || β(t)||dt = 1 0 ||q(t)|| 2 dt = 1 holds.
Therefore, the set of all SRVF functions associated with curves of length one are elements of a hypersphere in L 2 (since their norms are one). This greatly simplifies the shape analysis due to the fact that the differential geometry of a sphere is well-known. For example, if q 1 and q 2 are two elements of a unit hypersphere, the geodesic distance between them is given by length of shortest arc connecting them on the sphere. This length is actually given by cos -1 ( 2π 0 q 1 (t), q 2 (t) dt). As in the previous two cases, this distance does not depend on the re-parameterization of the two curves. That is, for any q 1 , q 2 and γ,

cos -1 ( 2π 0 q 1 (t), q 2 (t) dt) = cos -1 ( 2π 0 (q 1 , γ)(t), (q 2 , γ)(t) dt).
This leads to the definition of a distance between two curves that depends only on their shapes and orientations:

d c (β 1 , β 2 ) = min γ∈Γ cos -1 ( 2π 0 (q 1 , γ)(t), (q 2 , γ)(t) dt) .
(3) In this case, the geodesic path between the two curves is given by:

ψ(τ ) = 1 sin(θ) [sin(θ -τ θ)q 1 + sin(τ θ)(q 2 , γ * )] ,
where θ = d c (β 1 , β 2 ).

Comparison using shape and scale

As the next case, we are interested in comparing curves according to their shapes and scales, i.e. we want to remove the rigid motions from the representations. Let SO(3) the set of all possible rotations in R 3 . If we rotate the curve β by a rotation matrix O ∈ SO(3), we obtain the curve Oβ(t).

The SRVF of the rotated curve is given by Oq where q is the SRVF of the original curve. Consequently, the SRVF of a rotated and re-parameterized curve is given by γ(t)Oq(γ(t)).

To include the scale, we return to the SRVF representation of unscaled curves, and the distance function given in Eqn. 2 applies. The SRVFs are no longer restricted to a sphere but are elements of the full space. In order to remove the orientation feature, we need to add the minimization over SO(3) as follows. Define a distance function:

d d (β 1 , β 2 ) = min γ∈Γ,O∈SO(3) q 1 -O(q 2 , γ) 2 . (4)
Let γ * and O * be the re-parameterization and the rotation that minimize the right side in this equation. Then, the geodesic path between any two curves is given by:

ψ(τ ) = (1 -τ )q 1 + τ (O * q 2 , γ * ) .

Comparison using shape only

In the final case, we are interested in comparing curves only according to their shapes. That is, we want to remove all other physical variables (positions, scales, and orientations) from the representations, and want to consider only the influence of shapes in clustering and classification of fibers. The geodesic distance between any two scaled SRVFs is given by Eqn. 3. In order to remove orientation, we have to minimize over SO(3) as well as Γ now. Define a distance function:

de(β1, β2) = min γ∈Γ,O∈SO (3)
cos -1 (

Z 2π 0 (q1, γ)(t), O(q2, γ)(t) dt). (5)
Let γ * and O * be the re-parameterization and the rotation that minimize the right side in this equation. Then, the geodesic path between any two curves is given by: for θ = d e (β 1 , β 2 ),

ψ(τ ) = 1 sin(θ) [sin(θ -τ θ)q 1 + sin(τ θ)(O * q 2 , γ * )] .

CLUSTERING CORPUS CALLOSAL FIBERS

As an application, we consider the clustering of fibers of the corpus callosum using distances derived from the methods described above.

The corpus callosum(CC) is a large collection of fibers that connect the left and right hemispheres (Fig. 1). Pathologies such as multiple sclerosis [START_REF] Evangelou | Regional axonal loss in the corpus callosum correlates with cerebral white matter lesion volume and distribution in multiple sclerosis[END_REF], schizophrenia and Alzheimer's disease [START_REF] Head | Differential Vulnerability of Anterior White Matter in Nondemented Aging with Minimal Acceleration in Dementia of the Alzheimer Type: Evidence from Diffusion Tensor Imaging[END_REF] selectively affect specific regions of the CC and so there is interest in segregating the different regions for study. Clustering is made difficult both because there are no obvious landmarks and also because a limitation in DT-MRI tractography [START_REF] Ono | Atlas of the Cerebral Sulci[END_REF] renders fibers which have small differences in shape and scale along the length of the CC.

Combining shape, orientation and scale feature distances gives better discrimination than shape information alone. In the case of the CC, the genu(blue) and splenium(green) are at opposite ends and are easily grouped using the joint shape, orientation and scale metric, d b (Fig. 2a). The results for shape clustering alone (using d e ) are shown in Fig. 2b for comparison.

There are instances where a combination of shape, orientation and scale do not yield the best results. When a third bundle, the anterior section of the corpus(red) is added for clustering, the joint shape and orientation distance, d c , consistently gives better results than the d b metric (Fig. 3). This can be explained by the fact that the genu(blue) and corpus(red) have roughly the same scale and tend to cluster together with a d b metric. They are, however, oriented in different directions (0 • , 90 • , 180 • respectively) and so the d c metric, which does not include scale, provides better discrimination.

A second example is the clustering of the isthmus(magenta) and the splenium(green) at the posterior end of the CC(Fig. 4). The scale parameter, similar for the two groups, dominates the clustering and using either the shape and scale, d d , or the d b distance gives poor results. The d c metric, which does not include scale information, gives the best results (Fig. 4a).

CONCLUSION

This conceptual paper presents a comprehensive Riemannian framework for comparing fiber tracts using a variety of physical variables -shapes, positions, orientations, and scales. Depending on the application, some combination of these variables may be sufficient for a quantitative analysis. We consider a number of these combinations and for each combination we provide: (i) a geodesic distance function for comparing curves according to the chosen features, (ii) a geodesic path equation for computing geodesic paths between given curves. Both these quantities are invariant to the re-parameterizations of curves. This framework is versatile, in that, it allows for inclusion of any combination of shape, scale, orientation and translation in the analysis. The primary application of this framework analysis of 3D open curves is classification of DT-MRI white matter fibers in the brain. This framework can be extended to include additional features, e.g. the fractional anisotropy function and the mean diffusivity function along the fibers, to further strengthen the quantitative analysis [START_REF] Liu | Joint shape and texture analysis of objects boundaries in images using a Riemannian approach[END_REF].

Fig. 1 .

 1 Fig. 1. A mid-sagittal view of the corpus callosum. The rostrum and genu(blue), rostral body(cyan), anterior corpus(red), posterior corpus(black), isthmus(magenta), tapetum(yellow) and splenium(green) subdivisions are based on the Witelson [12] classification.

Fig. 2 .

 2 Fig.2. Clustering the genu and splenium, the anterior and posterior sections of the CC. Here, shape information alone (2b) is not adequate for clustering.

Fig. 3 .

 3 Fig. 3. Clustering of the genu, corpus and splenium, the anterior, middle and posterior sections of the CC. Including the scale information results in poorer clustering (3b).

Fig. 4 .

 4 Fig. 4. Clustering the isthmus and splenium, the posterior CC.


Xavier Bressaud
email: bressaud@iml.univ-mrs.fr

AND Fabien Durand
email: fabien.durand@u-picardie.fr

Alejandro Maass
email: amaass@dim.uchile.cl

EIGENVALUES OF FINITE RANK BRATTELI-VERSHIK DYNAMICAL SYSTEMS

Keywords: Mathematics Subject Classification. Primary: 54H20; Secondary: 37B20. Key words and phrases. minimal Cantor systems, finite rank Bratteli-Vershik dynamical systems, eigenvalues

In this article we study conditions to be a continuous or a measurable eigenvalue of finite rank minimal Cantor systems, that is, systems given by an ordered Bratteli diagram with a bounded number of vertices per level. We prove that continuous eigenvalues always come from the stable subspace associated to the incidence matrices of the Bratteli diagram and we study rationally independent generators of the additive group of continuous eigenvalues. Given an ergodic probability measure, we provide a general necessary condition to be a measurable eigenvalue. Then we consider two families of examples. A first one to illustrate that measurable eigenvalues do not need to come from the stable space. Finally we study Toeplitz type Cantor minimal systems of finite rank. We recover classical results in the continuous case and we prove measurable eigenvalues are always rational but not necessarily continuous.

Introduction

The study of eigenvalues of dynamical systems has been extensively considered in ergodic theory to understand and build the Kronecker factor and also to study the weak mixing property. In topological dynamics one also consider continuous eigenvalues, that is, eigenvalues associated to continuous eigenfunctions, to study topological weak mixing (at least in the minimal case). Since continuous eigenvalues are also eigenvalues, a recurrent question is to know whether they coincide. In general the answer is negative, since there are minimal topologically weakly mixing systems that are not weakly mixing for some invariant measure. A positive answer to that question has been given for the class of primitive substitution systems in [Ho]. The same question has been considered for linearly recurrent Cantor minimal systems, which contains substitution systems, in [CDHM] and [BDM] concluding that in general not all eigenvalues are continuous. Nevertheless, explicit necessary and sufficient conditions are given to check whether a complex number is an eigenvalue, continuous or not, that allow to recover the result in [Ho]. Those conditions only depend on the incidence matrices associated to a Bratteli-Vershik representation of the linearly recurrent minimal system and not on the partial order of the diagram. The independence of the order seems to be characteristic of linearly recurrent systems.

In this article we consider the same question for Cantor minimal systems that admit a Bratteli-Vershik representation with the same number of vertices per level. We call them (topologically) finite rank Cantor minimal systems. The motivations are different. First this class contains linearly recurrent systems but it is much larger and natural (some systems in this class appear as the symbolic representation of well studied classes of dynamical systems like interval exchange transformations [GJ]). Second, the knowledge about this class of systems is very small, one of its main (and recent) properties is that they are either expansive or equicontinuous [DM]. Thus, up to odometers that are well known, this is a huge class of symbolic minimal systems.

In Section 3 we study continuous eigenvalues. The main result states that continuous eigenvalues of Cantor minimal systems always come from the stable space associated to the sequence of matrices of the Bratteli-Vershik representation of the system and we provide a general necessary condition to be a continuous eigenvalue. In Section 4 these results are used to get a bound for the maximal number of rationally independent continuous eigenvalues of a finite rank system. In section 5 we give a necessary condition to be an eigenvalue of a finite rank minimal systems indowed with an invariant probability measure. Next two sections are devoted to examples. In section 6 we construct a uniquely ergodic non weakly mixing finite rank Cantor minimal system to illustrate that eigenvalues do not always come from the stable space associated to the sequence of matrices given by the Bratteli-Vershik representation of the system. In the last section we study Toeplitz type Cantor minimal systems of finite rank. The main property we deduce is that eigenvalues are always rational but not always continuous.

Basic definitions

2.1. Dynamical systems and eigenvalues. A topological dynamical system, or just dynamical system, is a compact Hausdorff space X together with a homeomorphism T : X → X. One uses the notation (X, T ). If X is a Cantor set one says that the system is Cantor. That is, X has a countable basis of closed and open sets and it has no isolated points. A dynamical system is minimal if all orbits are dense in X, or equivalently the only non trivial closed invariant set is X. A complex number λ is a continuous eigenvalue of (X, T ) if there exists a continuous function f : X → C, f = 0, such that f • T = λf ; f is called a continuous eigenfunction (associated to λ). Let µ be a T -invariant probability measure, i.e., T µ = µ, defined on the Borel σ-algebra B(X) of X. A complex number λ is an eigenvalue of the dynamical system (X, T ) with respect to µ if there exists f ∈ L 2 (X, B(X), µ), f = 0, such that f • T = λf ; f is called an eigenfunction (associated to λ). If the system is ergodic, then every eigenvalue is of modulus 1, and every eigenfunction has a constant modulus µ-almost surely. Of course continuous eigenvalues are eigenvalues.

2.2. Bratteli-Vershik representations. Let (X, T ) be a Cantor minimal system. It can be represented by an ordered Bratteli-Vershik diagram. For details on this theory see [HPS]. We give a brief outline of such constructions emphasizing the notations used in the paper.

2.2.1. Bratteli-Vershik diagrams. A Bratteli-Vershik diagram is an infinite graph (V, E) which consists of a vertex set V and an edge set E, both of which are divided into levels

V = V 0 ∪ V 1 ∪ . . ., E = E 1 ∪ E 2 ∪ .
. . and all levels are pairwise disjoint. The set V 0 is a singleton {v 0 }, and for k ≥ 1, E k is the set of edges joining vertices in V k-1 to vertices in V k . It is also required that every vertex in V k is the "end-point" of some edge in E k for k ≥ 1, and an "initial-point" of some edge in E k+1 for k ≥ 0. One puts V k = {1, . . . , C(k)}. The level k is the subgraph consisting of the vertices in V k ∪ V k+1 and the edges E k+1 between these vertices. Level 0 is called the hat of the Bratteli-Vershik diagram and it is uniquely determined by an integer vector 1) , where each component represents the number of edges joining v 0 and a vertex of V 1 . We will assume H(1) = (1, . . . , 1) T ; it is not restrictive for our purpose. We describe the edge set

H(1) = h 1 (1), . . . , h C(1) (1) T ∈ N C(
E k using a V k × V k-1 incidence matrix M (k) for which its (i, j)-entry is the number of edges in E k joining vertex j ∈ V k-1 with vertex i ∈ V k .
We also assume M (k) > 0 (this is not a restriction for our purpose). For 1 ≤ k ≤ l one defines

P (k) = M (k) • • • M (2) and P (l, k) = M (l) • • • M (k + 1)
with P (1) = I and P (k, k) = I ; where I is the identity map. Also, put

H(k) = P (k)H(1) = (h 1 (k), . . . , h C(k) (k)) T . 2.2.2. Ordered Bratteli-Vershik diagrams. An ordered Bratteli-Vershik diagram is a triple B = (V, E,
) where (V, E) is a Bratteli-Vershik diagram and a partial ordering on E such that : Edges e and e ′ are comparable if and only if they have the same end-point. Let k ≤ l in N \ {0} and let E k,l be the set of all paths in the graph joining vertices of V k-1 with vertices of V l . The partial ordering of E induces another in E k,l given by (e k , . . . , e l ) (f k , . . . , f l ) if and only if there is k ≤ i ≤ l such that e j = f j for i < j ≤ l and e i f i . Given a strictly increasing sequence of integers (m n ) n≥0 with m 0 = 0 one defines the contraction of B = (V, E, ) (with respect to (m n ) n≥0 ) as

(V mn ) n≥0 , E mn+1,mn+1 n≥0 , ,
where is the order induced in each set of edges E mn+1,mn+1 . The converse operation is called miscroscoping (see [HPS] for more details). Given an ordered Bratteli-Vershik diagram B = (V, E, ) one defines X B as the set of infinite paths (e 1 , e 2 , • • • ) starting in v 0 such that for all i ≥ 1 the end-point of e i ∈ E i is the initial-point of e i+1 ∈ E i+1 . We topologize X B by postulating a basis of open sets, namely the family of cylinder sets

[e 1 , e 2 , . . . , e k ] = { (x 1 , x 2 , . . .) ∈ X B | x i = e i , for 1 ≤ i ≤ k } .
Each [e 1 , e 2 , . . . , e k ] is also closed, as is easily seen, and so we observe that X B is a compact, totally disconnected metrizable space. When there is a unique x = (x 1 , x 2 , . . .) ∈ X B such that x i is maximal for any i ≥ 1 and a unique y = (y 1 , y 2 , . . .) ∈ X B such that y i is minimal for any i ≥ 1, one says that B = (V, E, ) is a properly ordered Bratteli diagram. Call these particular points x max and x min respectively. In this case one defines a dynamic V B over X B called the Vershik map. The map V B is defined as follows: let x = (x 1 , x 2 , . . .) ∈ X B \ {x max } and let k ≥ 1 be the smallest integer so that x k is not a maximal edge. Let y k be the successor of x k and (y 1 , . . . , y k-1 ) be the unique minimal path in E 1,k-1 connecting v 0 with the initial point of y k . One sets V B (x) = (y 1 , . . . , y k-1 , y k , x k+1 , . . .) and V B (x max ) = x min .

The dynamical system (X B , V B ) is called the Bratteli-Vershik system generated by B = (V, E, ). The dynamical system induced by any contraction of B is topologically conjugate to (X B , V B ). In [HPS] it is proved that any Cantor minimal system (X, T ) is topologically conjugate to a Bratteli-Vershik system (X B , V B ). One says that (X B , V B ) is a Bratteli-Vershik representation of (X, T ). In the sequel we identify (X, T ) with one of its Bratteli-Vershik representations. We always assume the representations are proper. A Cantor minimal system is of (topological) finite rank d ≥ 1 if it admits a Bratteli-Vershik representation such that the number of vertices per level verify C(k) ≤ d for any k ≥ 1. Contracting and microscoping the diagram if needed one can assume (this is done in the sequel) that C(k) = d for any k ≥ 2. A Cantor minimal system is linearly recurrent if it admits a Bratteli-Vershik representation such that the set {M (n); n ≥ 1} is finite. Clearly linearly recurrent Cantor minimal systems are of finite rank (for details about these systems see [START_REF] Durand | Linearly recurrent subshifts have a finite number of non-periodic subshift factors[END_REF]).

2.2.3. Associated Kakutani-Rohlin partitions and invariant measures. Let (X, T ) be the minimal Cantor system defined by a properly ordered Bratteli diagrams B = (V, E, ). The ordered Bratteli diagram defines for each n ≥ 0 a clopen Kakutani-Rohlin (KR) partition of X

P(n) = {T -j B k (n); k ∈ V n , 0 ≤ j < h k (n)} , with B k (n) = [e 1 , . . . e n ],
where (e 1 , . . . , e n ) is the unique path from v 0 to k such that each e i is minimal for the ordering of B. For each k ∈ V n the set {T -j B k (n); 0 ≤ j < h k (n)} is the k-th tower of P(n). This corresponds to the set of all the paths from v 0 to k ∈ V n (there are exactly h k (n) such paths). The map τ n : X → V n is given by τ n (x) = k if x belongs to the k-th tower of P(n). Denote by T n the σ-algebra generated by partition the P(n); that is, the finite paths joining v 0 with any vertex of V n . Let µ be a T -invariant measure. It is determined by its value in B k (n) for each n ≥ 0 and k ∈ V n . Define µ(n) = (µ 1 (n), . . . , µ C(n) ) T with µ k (n) = µ(B k (n)). A simple computation yields to the following fundamental relation:

µ(n) = M T (n + 1)µ(n + 1) for any n ≥ 1. 2.2.4. Return and suffix maps. Fix n ∈ N. The return time of x to B τn(x) (n) is given by r n (x) = min{j ≥ 0; T j x ∈ B τn(x) (n)}. Define the map s n : X → N C(n) , called suffix map of order n, by (s n (x)) k = |{e ∈ E n+1 ;
x n+1 e, x n+1 = e, k is the initial vertex of e}| for all x = (x n ) ∈ X and k ∈ V n . A classical computation gives (see for example [BDM])

r n (x) = s 0 (x) + n-1 k=1 s k (x), P (k)H(1) . (2.1)

Continuous eigenvalues : general necessary and sufficient conditions

Let (X, T ) be a Cantor minimal system given by a Bratteli-Vershik representation B = (V, E, ). Recall the associated sequence of matrices is (M (n); n ≥ 1), M (n) > 0 and

P (n) = M (n) • • • M (2)
for n ≥ 2, and P (1) = I. First we recall a general necessary and sufficient condition to be a continuous eigenvalue of (X, T ) proved in [BDM].

Proposition 1. Let λ = exp(2iπα). The following conditions are equivalent, (1) λ is a continuous eigenvalue of the Cantor minimal system (X, T );

(2) (λ rn(x) ; n ≥ 1) converges uniformly in x, i.e., the sequence (αr n (x); n ≥ 1) converges (mod Z) uniformly in x.

It follows that,

Corollary 2. Let λ ∈ S 1 = {z ∈ C; |z| = 1}. If λ is a continuous eigenvalue of (X, T ) then lim n→∞ λ hj n (n) = 1 uniformly in (j n ; n ∈ N) ∈ n∈N {1, . . . , C(n)}.
The following theorem states that the necessary condition in Theorem 1 part (2) in [BDM] is true in general. Denote by ||| • ||| the distance to the nearest integer vector.

Theorem 3. Let (X, T ) be a Cantor minimal system given by a Bratteli-Vershik representation

B = (V, E, ). If λ = exp(2iπα) is a continuous eigenvalue of (X, T ) then n≥1 |||αP (n)H(1)||| < ∞ .
Before proving Theorem 3 we introduce an intermediate statement which gives a more precise interpretation to this necessary condition.

Lemma 4. Assume coefficients of matrices (M (n); n ≥ 1) are strictly bigger than 1. Let (j(n); n ∈ N) and (i(n); n ∈ N) be sequences of positive integers such that j(n + 1) -j(n) ≥ 2 and i(n) ∈ {1, . . . , C(j(n))} for all n ∈ N. Then there exist different points x and y in X such that:

(1) ∀ n ∈ N, s j(n) (x) -s j(n) (y) = can i(n) (the i(n)-th canonical vector);

(2) s j (x) -s j (y) = (0, . . . , 0) T whenever j ∈ {j(n); n ∈ N}.

Proof. For all n ∈ N we choose α(n) ∈ V (n). We defined x = (e(n)) n∈N and y = (f (n)) n∈N and

Proof of Theorem 3. We assume without loss of generality that the coefficients of the matrices (M (n); n ≥ 1) are strictly bigger than 1 and that H(1) = (1, . . . , 1) T . Let λ be a continuous eigenvalue of (X, T ). We deduce from Corollary 2 that |||αP (n)H(1)||| converges to 0 as n → ∞. Then for all n ≥ 1 there exist a real vector v(n) and an integer vector w(n) such that

αP (n)H(1) = v(n) + w(n) and v(n) → n→∞ 0 .
Thus it is enough to prove that n≥1 v(n) < ∞.

For n ≥ 1 let l(n) ∈ {1, ..., C(n)} be such that

| e l(n) , v(n) | = max l∈{1,...,C(n)} | e l , v(n) |
where e l is the l-th canonical vector of R C(n) . Let

I + = {n ≥ 1; e l(n) , v(n) ≥ 0}, I -= {n ≥ 1; e l(n) , v(n) >< 0}.
To prove n≥1 v(n) < ∞ one only needs to show that

n∈I + e l(n) , v(n) < ∞ and - n∈I - e l(n) , v(n) < ∞.
Since the arguments we will use are similar in both cases we only prove the first one. Moreover, to prove

n∈I + e l(n) , v(n) < ∞ we only show n∈I + ∩2N e l(n) , v(n) < ∞. Analogously, one can prove that n∈I + ∩(2N+1) e l(n) , v(n) < ∞.
Assume I + ∩ 2N is infinite, if not the result follows directly. Order its elements j(0) < j(1) < . . . < j(n) < . . . and define i(n) = l(j(n)) for n ∈ N. Let x, y ∈ X be the points given by Lemma 4 using these sequences. Now, from equality (2.1) and Proposition 1, one deduces that

α(r m (x) -r m (y)) = α n∈{1,...,m-1}∩I + ∩2N (s n (x) -s n (y)), P (n)H(1) + (s 0 (x) -s 0 (y)) = α n∈{1,...,m-1}∩I + ∩2N e l(n) , P (n)H(1) = n∈{1,...,m-1}∩I + ∩2N e l(n) , v(n) + w(n) converges ( mod Z) when m → ∞. Then n∈{1,...,m-1}∩I + ∩2N e l(n) , v(n) con- verges ( mod Z) when m → ∞. But e l(n) , v(n) tends to 0, hence the series n∈I + ∩2N e l(n) , v(n) converges.
The following theorem states that continuous eigenvalues are always constructed from the subspaces,

v ∈ R C(m) ; P (n, m)v → n→∞ 0 , m ≥ 2,
defined by the sequence of incidence matrices (M (n); n ≥ 1). In the sequel we use the norm ||.|| defined by ||v|| = max i |v i | for all v ∈ R d .

Theorem 5. Let (X, T ) be a Cantor minimal system given by a Bratteli-Vershik representation

B = (V, E, ). Let λ = exp(2iπα) be a continuous eigenvalue of (X, T ). Then, there exist m ∈ N, v ∈ R C(m) and w ∈ Z C(m) such that αP (m)H(1) = v + w and P (n, m)v → n→∞ 0.
Proof. We deduce from Corollary 2 that |||αP (n)H(1)||| converges to 0 as n tends to ∞. Hence, for every n ≥ 2 one can write αP

(n)H(1) = v(n) + w(n), where w(n) is an integer vector and v(n) is a real vector with v(n) → 0 as n → ∞. Clearly αP (n + 1)H(1) = M (n + 1)v(n) + M (n + 1)w(n) = v(n + 1) + w(n + 1). (3.1)
We start proving that there exists m ≥ 1 such that for all n ≥ m one has M (n + 1)v(n) = v(n + 1). From Proposition 1 one knows there exists m ≥ 1 such that for all n ≥ m and all x ∈ X it holds

||| s n (x), αP (n)H(1) ||| < 1 4 and v(n) < 1 4 .
Hence, for all n ≥ m and all x ∈ X, one gets

||| s n (x), v(n) ||| < 1 4 . (3.2) Fix n ≥ m. Consider x ∈ B k (n + 1) for some 1 ≤ k ≤ C(n + 1) and let 0 = j 1 < j 2 < • • • < j l be the collection of all the integers 0 ≤ j < h k (n + 1) such that T -j x ∈ ∪ i∈{1,...,C(n)} B i (n). Remark that s T n (T -j l x) -e T k M (n + 1) = 1 (3.3) s n (T -jm+1 x) -s n (T -jm x) = 1 (3.4) for all 1 ≤ m ≤ l -1. Let 1 ≤ m ≤ l -1 and suppose s n (T -jm x), v(n) < 1/4. Then, from (3.4), | s n (T -jm+1 x), v(n) | = | s n (T -jm x), v(n) + s n (T -jm+1 x), v(n) -s n (T -jm x), v(n) | < 1 2 . From (3.2) one gets that | s n (T -jm+1 x), v(n) | < 1 4 . Thus, as | s n (x), v(n) | = 0, it follows by induction that s n (T -j l x), v(n) < 1 4 . Therefore, from (3.3) one deduces that | e k , M (n + 1)v(n) | < 1/2. This is true for all 1 ≤ k ≤ C(n + 1), then M (n + 1)v(n) < 1/2.
Finally, from (3.1) one deduces that for all n ≥ m,

M (n + 1)v(n) = v(n + 1) and M (n + 1)w(n) = w(n + 1). (3.5)
To conclude it is enough to set v = v(m) and w = w(m). Lemma 6. Let (X, T ) be a Cantor minimal system given by a Bratteli-Vershik representation

B = (V, E, ). Consider m ∈ N and v ∈ R C(m) such that P (n, m)v → 0 as n → ∞. Then v, µ(m) = 0 for any T -invariant probability measure µ. Proof. From definition one has v, µ(m) = v, P T (n, m)µ(n) = P (n, m)v, µ(n) ≤ P (n, m)v • µ(n)
and the last term converges to 0 as n → ∞. Thus v, µ(m) = 0.

Corollary 7. Let (X, T ) be a Cantor minimal system given by a Bratteli-Vershik representation B = (V, E, ) and let µ be a T -invariant probability measure. Let λ = exp(2iπα) be a continuous eigenvalue of (X, T ). Then one of the following conditions holds:

(1) α is rational with a denominator dividing gcd(h i (m); 1 ≤ i ≤ C(m)) for some m ∈ N.

(2) There exist m ∈ N and an integer vector w ∈ Z C(m) such that α = w, µ(m) .

Proof. Let m, v and w be as in Theorem 5. We recall that P (m)H(1) = H(m).

Assume v = 0. Then αH(m) = w and thus α is rational with a denominator dividing gcd(h i (m);

1 ≤ i ≤ C(m)). Now suppose v = 0. From Lemma 6, v, µ(m) = 0. Thus, from w = αH(m) -v and µ(m), H(m) = 1 one gets α = w, µ(m) .
Part (2) of previous lemma left open the question whether any integer vector w ∈ Z C(m) , for some m ∈ N, can produce a continuous eigenvalue of the system by taking α = w, µ(m) . It is enough to consider topological weakly mixing Cantor minimal systems to see that in some cases not all integer vectors can produce a continuous eigenvalue. In general, the set of integer vectors from which one can define continuous eigenvalues of the system is a discrete group. Normally, very difficult to describe explicitly. In the next section we give a slightly more precise description of such group in the finite rank case.

Continuous eigenvalues of finite rank systems

Let (X, T ) be Cantor minimal system of finite rank d. Fix a Bratteli-Vershik representation of (X, T ) with exactly d vertices per level which sequence of incidence matrices is (M (n); n ≥ 1).

4.1. Rationally independent continuous eigenvalues. Let E(X, T ) be the additive group of continuous eigenvalues of (X, T ), that is,

E(X, T ) = {α ∈ R; exp(2iπα) is a continuous eigenvalue of (X, T )} .
In this section we study the maximal number η(X, T ) of rationally independent elements in E(X, T ). Remark that 1 is always an eigenvalue of (X, T ) so Z ⊆ E(X, T ). We need the following lemma whose proof is left to the reader.

Lemma 8. Let (X, T ) be a Cantor minimal system of finite rank d. Then, there are at most d ergodic measures µ 1 , . . . , µ l (l ≤ d). Moreover, there exits m ∈ N such that for all n ≥ m vectors µ 1 (n), . . . , µ l (n) are linearly independent.

Theorem 9. Let (X, T ) be a Cantor minimal system of finite rank d. Let µ 1 , . . . , µ l , l ≤ d, be all its ergodic measures. Then, η(X, T ) ≤ d -l + 1.

Proof. Fix a Bratteli-Vershik representation of (X, T ) with exactly d vertices per level. Put η = η(X, T ) and assume η > d -l + 1. Let {α 1 , . . . , α η } be a set of rationally independent elements in E(X, T ). From Theorem 5, there exist m ∈ N and vectors

v i ∈ R d , w i ∈ Z d , for i ∈ {1, . . . , η}, such that α i H(m) -v i = w i and P (n, m)v i → n→∞ 0.
Consider m so large that Lemma 8 is also verified from such an integer. From Lemma 6 one has that for all 1 ≤ i ≤ η and all 1

≤ j ≤ l, w i , µ j (m) = α i . Thus, w i , µ 1 (m)-µ j (m) = 0 for 2 ≤ j ≤ l. Now, from Lemma 8 one deduces that {µ 1 (m) -µ 2 (m), . . . , µ 1 (m) -µ l (m)} generates a (l -1)-dimensional vector space.
We conclude that the linear space generated by w 1 , . . . , w η is of dimension at most d-l+1. Consequently, there exist integers ǫ 1 , . . . , ǫ d-l+2 with |ǫ 1 |+. . .+|ǫ d-l+2 | = 0 and

ǫ 1 w 1 + . . . + ǫ d-l+2 w d-l+2 = 0 . Thus, ǫ 1 α 1 + . . . + ǫ d-l+2 α d-l+2 = 0
which contradicts the fact that {α 1 , . . . , α η } is a set of rationally independent elements in E(X, T ).

Remark that from the proof of the theorem, a set of rationally independent generators of E(X, T ) can be determined from a single level m once we know µ(m) and good integer vectors. Of course, this level m can be very large and difficult to get.

Put η = η(X, T ). If η = d we say that (X, T ) is of maximal type. From Theorem 9 one has that maximal type systems are uniquely ergodic. Conversely, if (X, T ) is uniquely ergodic, then it has at most d rationally independent continuous eigenvalues but it is not necessarily of maximal type. As an example consider a substitution system whose incidence matrix M (n) = A for all n ≥ 2 such that A is primitive and has two real eigenvalues with modulus bigger than one (for details about Bratteli-Vershik representations of substitution systems see [DHS]). Since 1 ∈ E(X, T ), one can always produce rationally independent generators of E(X, T ) containing 1. Observe that rational eigenvalues are associated to 1. Fix {1, α 1 , . . . , α η-1 } a set of rationally independent generators of E(X, T ). Let µ be an ergodic measure of (X, T ).

From Theorem 5 there is m ∈ N such that for all 1 ≤ i ≤ η -1 there exist a real vector v i ∈ R d and an integer vector

w i ∈ Z d satisfying α i H(m) = v i + w i and P (n, m)v i → 0 as n → ∞.
From Lemma 6, each v i is orthogonal to the linear space spanned by µ(m), thus (4.1)

α i = w i , µ(m) .
Proposition 10. The vectors {v 1 , . . . , v η-1 } and the vectors {w 1 , . . . , w η-1 , H(m)} are linearly independent.

Proof. Suppose η-1 i=1 δ i w i = 0 with some δ i = 0. Since w 1 , . . . , w η-1 are integer vectors we can assume the δ i 's are integer numbers. From α i = w i , µ(m) for all 1 ≤ i ≤ η -1 one gets η-1 i=1 δ i α i = 0 with some δ i = 0. But α 1 , . . . , α η-1 are rationally independent, then coefficients δ i 's must be 0, a contradiction. Then w 1 , . . . , w η-1 are linearly independent. Now, it holds that H(m) / ∈ {w 1 , . . . , w η-1 } . Indeed, if H(m) = η-1 j=1 q j w j , with rational coefficients, then by taking the inner product with µ(m) one gets that 1 = η-1 j=1 q j α j . This contradicts the fact that 1, α 1 , . . . , α η-1 are rationally independent. One concludes w 1 , . . . , w η-1 , H(m) are linearly independent. Therefore, from

η-1 j=1 λ j v j = 0 one deduces ( η-1 j=1 λ j α j )H(m) - η-1 j=1 λ j w j = 0 and thus λ 1 = . . . = λ η-1 = 0.
Fix an ergodic measure µ and, for each n ≥ 1, define ζ(µ, n) to be the maximal number of rationally independent components of µ(n).

Proposition 11. For all n ≥ m, ζ(µ, n) ≥ η. In particular, if the system is of maximal type, then ζ(µ, n) = d for n ≥ m.
Proof. We give a proof for n = m, for a general n it is analogous. Let q = (q 1 , . . . , q d ) T ∈ Q d be such that q, µ(m) = 0. Thus q is not contained in the linear space W. Indeed, if q = η-1 i=1 r i w i + rH(m) with r and the r i 's rational numbers, then, taking the inner-product with µ(m), one obtains 0 = η-1 i=1 r i α i + r which implies r 1 = . . . = r η-1 = r = 0 (recall 1, α 1 , . . . , α η-1 are rationally independent). Assume for all subsets J of {1, . . . , d} with cardinality η there is a non zero rational vector q J ∈ Q d with q J j = 0 for j ∈ {1, . . . , d} \ J such that q J , µ(m) = 0. At least d -η + 1 of such vectors must be linearly independent. To prove this fact consider the family J i = {i, . . . , i + η -1} for i ∈ {1, . . . , d -η + 1} and the corresponding vectors q J1 , . . . , q J d-η+1 . From the first part of the proof one concludes that H(m), w 1 , . . . , w η-1 , q J1 , . . . , q J d-η+1 are d + 1 independent vectors in R d , which is a contradiction. Therefore, there is J ⊆ {1, . . . , d} with cardinality η such that µ j (m), j ∈ J, are rationally independent components of µ(m). This gives ζ(µ, n) ≥ η. The maximal type case follows directly.

Observe that the inequality in the proposition can be strict. 4.2. Dimension group and geometric interpretation of eigenvalues. Observe that it is not enough to have v ∈ R d with P (n, m)v → 0 as n → ∞ and w ∈ Z d such that v + w = αH(m) for some m ≥ 1 to ensure exp(2iπα) is a continuous eigenvalue of (X, T ). In addition, from Proposition 1, it is also necessary that the series n≥m s n (x), P (n, m)v converges modulo Z. In the next two propositions we try to give a more precise statement involving the so called dimension group associated to the sequence of matrices (M (n); n ≥ 1). Fix m ≥ 1 and an invariant measure µ. Put

V(m) = {µ(m)} ⊥ . Let V s (m) be the subspace of R d that is asymptotically contracted by (M (n); n ≥ m) : V s (m) = {v ∈ R d ; P (n, m)v → 0 as n → ∞}.
Also distinguish the subspaces of V s (m),

V 0 (m) = v ∈ R d ; ∃n ≥ m, P (n, m)v = 0 = n≥m Ker(P (n, m)) and V 1 (m) =    v ∈ R d ; n≥m P (n, m)v < ∞    . Obviously, V 0 (m) ⊆ V 1 (m) ⊆ V s (m) ⊆ V(m) .
One has P (m)V(1) ⊆ V(m). Equality holds if the matrices (M (n); n ≥ 1) are invertible.

Proposition 12. There exist m ≥ 1 and a linear space

V (m) ⊆ V(m) such that P (n, m) : V (m) → V(n) is one to one for any n > m.
Proof. Let us choose a subspace V (1) of R d such that V s (1) = V (1) ⊕ V 0 (1). Let m ≥ 1 and assume subspaces V (n) are defined for all 1 ≤ n ≤ m such that: m+1) . This procedure defines recursively a sequence of subspaces verifying for all m ≥ 2 and all 1 m+1) . Since P (n, m)V (m) ⊆ V (n) and in view of the definition of V 0 (m), P (n, m) is injective on V (m) , then the sequence (dim(V (m) ); n ≥ 1) is increasing. Since it is bounded by d, there is m ∈ N such that for all n ≥ m, P (n, m)V (m) = V (n) . This concludes the proof.

V s (n) = V 0 (n) ⊕ V (n) and P (n, k)V (k) ⊆ V (n) for all 1 ≤ k < n ≤ m. Choose a subspace W (m+1) of V s (m + 1) such that V s (m + 1) = V 0 (m + 1)⊕M (m+ 1)V (m) ⊕W (m+1) and set V (m+1) = M (n+1)V (m) ⊕W (
≤ k ≤ m, V s (m) = V 0 (m) ⊕ V (m) and P (m + 1, k)V (k) ⊆ V (
Fix the integer m found in the previous proposition. Notice that if the matrices (M (n); n ≥ 1) are invertible, then one can take m = 1.

Consider the discrete subgroup of R d G(m) = n≥m P (n, m) -1 Z d = z ∈ Q d ; ∃n ≥ m, P (n, m)z ∈ Z d and the one dimensional subspace ∆(m) = {tH(m); t ∈ R} ⊆ R d . Proposition 13. Let λ = exp(2iπα). If λ is a continuous eigenvalue of (X, T ) then αH(m) ∈ (G(m) + V 1 (m)) ∩ ∆(m).
Proof. According to Theorem 3,

n≥1 |||αP (n)H(1)||| < ∞ .
From Lemma 6, there exist m ′ ≥ 1, an integer vector w ′ ∈ Z d and a real vector

v ′ ∈ R d with αH(m ′ ) = v ′ + w ′ and P (n, m ′ )v ′ → 0 as n → ∞ . One can assume m ′ ≥ m. Since v ′ ∈ V s (m ′ ), it splits into v ′ = v ′ 0 + v ′ s , with v ′ 0 ∈ V 0 (m ′ ) and v ′ s ∈ V (m ′ ) . There is v ∈ V (m) such that v ′ s = P (m ′ , m)v. Hence, αP (m ′ , m)H(m) = w ′ + v ′ 0 + P (m ′ , m)v.
Let n be such that P (n, m ′ )v ′ 0 = 0. One has, αP (n, m)H(m) = P (n, m ′ )w ′ + 0 + P (n, m)v.

One deduces that

P (n, m) (αH(m) -v) ∈ Z d , which means that αH(m) -v ∈ G(m) + Ker(P (n, m)). To conclude, notice that, since for n large enough, |||αP (n)H(1)||| = |||P (n, m)v||| = P (n, m)v , v must belong to V 1 (m), so that αH(m) ∈ (V 1 (m) + G(m)) ∩ ∆(m).
Remark 14. G(m) is associated to the so called dimension group. It is classically presented as a quotient G ′ = H/ ∼, where

H = (z, p) ∈ Q d × N; ∃n ≥ p, P (n, p)z ∈ Z d
and (z, p) ∼ (y, q) ⇔ ∃n ≥ p, n ≥ q, P (n, p)z = P (n, q)y. If the matrices (M (n); n ≥ 1) are invertible each g ∈ G ′ can be represented by the unique element (z, 1) ∈ H in class g. In the general case, some elements of G ′ do not have a representative of this type. Nevertheless, by previous propositions one can choose an appropriate m ≥ 1 and identify each g ∈ G ′ with a representative of the form (z, m) ∈ H if it is not asymptotically null, and with (0, m) if it is asymptotically null. The difference here is that two elements of G(m) may correspond to the same element of G ′ if their images coincide after a while. One has that G ′ is isomorphic to G(m)/ ≈ with z ≈ y ⇔ ∃n ≥ m, P (n, m)z = P (n, m)y.

Fix m ≥ 1 as before and such that 1, α 1 , . . . , α η-1 is a base of rationally independent continuous eigenvalues of (X, T ) with

α i H(m) = v i + w i , v i ∈ V s (m) and w i ∈ Z d . When ζ(µ, m) = d, the eigenvalues can be described from W = {w 1 , . . . , w η-1 , H(m)} . Proposition 15. Assume ζ(µ, m) = d (in particular if (X, T ) is of maximal type). Consider α = q + η-1 i=1 q i α i ∈ E(X, T ) with q, q 1 , . . . , q η-1 ∈ Q. Then qH(m) + η-1 i=1 q i w i belongs to G(m). Moreover, if α is a rational continuous eigenvalue then αH(m) belongs to G(m). Conversely, if q, q 1 , . . . , q η-1 ∈ Q are such that qH(m) + η-1 i=1 q i w i ∈ G(m) then α = q + η-1 i=1 q i α i ∈ E(X, T ).
Proof. Take α ∈ E(X, T ) as in the statement of the proposition. By Proposition 13, there are v ′ ∈ V s (m) and

w ′ ∈ G(m) such that αH(m) = v ′ + w ′ . Thus, v -v ′ = w ′ -w, where v = η-1 i=1 q i v i and w = qH(n) + η-1 i=1 q i w i . From v -v ′ , µ(m) = 0 one deduces that w -w ′ , µ(m) = 0. But ζ(µ, m) = d, thus w = w ′ and consequently v = v ′ . This proves the first result. If α ∈ Q then v = 0, which proves the second result. Now consider q, q 1 , . . . , q η-1 ∈ Q such that qH(m) + η-1 i=1 q i w i ∈ G(m). Put v = η-1 i=1 q i v i .
The series n≥m s n (x), P (n, m)v converges uniformly in x modulo Z, because the corresponding series with v i instead of v does. This proves α = q + η-1 i=1 q i α i belongs to E(X, T ). Define the matrix W = [w 1 , . . . , w η-1 , H(m)]. From (4.1), it is direct that

W T µ(m) = (α 1 , . . . , α η-1 , 1) T . Corollary 16. If ζ(µ, m) = d (in particular if (X, T ) is of maximal type) then E(X, T ) is isomorphic (as a group) with the discrete subgroup of Q d , Q(X, T ) = {z ∈ Q d ; W T z ∈ G(m)}.

Measurable eigenvalues of finite rank systems: a general necessary condition

Let (X, T ) be a Cantor minimal system of finite rank d and µ a T -ergodic measure. By contracting the associated Bratteli-Vershik diagram one can always assume there exists I ⊆ {1, . . . , d} such that:

(1) For all k ∈ I, lim inf n→∞ µ{τ n = k} > 0;

(2) For all k ∈ I c , n≥1 µ{τ n = k} < ∞.

A diagram verifying these properties will be called clean. From conditions (1) and

(2) one deduces that τ n (x) belongs to I from some n for almost all x ∈ X. Consider a measurable eigenfunction f : X → C of (X, T, µ) associated to the eigenvalue λ = exp(2iπα) with |f | = 1, µ-almost surely. For n ≥ 1 put

f n = E(f |T n ) = d k=1 h k (n)-1 j=0 1 T -j B k (n) 1 µ k (n) B k (n) λ -j f dµ and set 1 µ k (n) B k (n) f dµ = c k (n)λ ρ k (n) ,
with c k (n) ≥ 0. If x belongs to the k-th tower of level n one has that f n (x) = λ -rn(x)+ρ k (n) c k (n). A simple computation yields to the following interesting relation that we will not exploit in this article:

c l (n)µ l (n) ≤ d k=1 M k,l (n + 1)µ k (n + 1)c k (n + 1)
for all l ∈ {1, . . . , d}.

Lemma 17. For any

1 ≤ k ≤ d such that lim inf n→∞ µ{τ n = k} > 0 one has c k (n) → 1 as n → ∞. Proof. By construction, ||f n || 2 2 = d k=1 µ{τ n = k}c k (n) 2 → 1 as n → ∞. Since the c k (n) are bounded by one, then c k (n) → 1 as n → ∞ for each 1 ≤ k ≤ d such that lim inf n→∞ µ{τ n = k} > 0. For n ≥ 1 and k, l ∈ {1, . . . , d} define S n (l, k) = {s n (x); x ∈ X, τ n (x) = l, τ n+1 (x) = k}.
Proposition 18. Let (X, T ) be a Cantor minimal system of finite rank d and µ a Tergodic measure. Assume (X, T ) is given by a clean Bratteli-Vershik representation. If λ = exp(2iπα) is an eigenvalue of (X, T, µ), then for n ≥ 1 there exist real numbers ρ 1 (n), . . . , ρ d (n) such that the following series converges,

n≥1 max (l,k)∈J 1 M k,l (n + 1) s∈Sn(l,k) |1 -λ s,H(n) -ρ k (n+1)+ρ l (n) | 2 (5.1)
where J = {(l, k) ∈ {1, . . . , d} 2 ; lim inf n→∞ µ{τ n = l, τ n+1 = k} > 0}.

Proof. Let I be a subset of {1, . . . , d} verifying (1) and (2) in the definition of a clean Bratteli-Vershik representation. Let f : X → S 1 be an eigenfunction for the eigenvalue λ = exp(2iπα). As above, for n ≥ 1 and k ∈ {1, . . . , d}, we set

f n = E(f |T n ) and 1 µ k (n) B k (n) f dµ = c k (n)λ ρ k (n) with c k (n) ≥ 0. From Lemma 17, c k (n) → 1 as n → ∞ if k ∈ I. Let l, k ∈ J. Observe that l, k ∈ I too. One has b M k,l (n + 1) ≤ µ{τ n = l, τ n+1 = k} M k,l (n + 1) = M k,l (n + 1)h l (n)µ k (n + 1) M k,l (n + 1) (5.2) = h l (n)µ k (n + 1) , (5.3)
for some b > 0. On the other hand, since the sequence (f n ; n ≥ 1) is a martingale, then n≥1 ||f n+1 -f n || 2 2 converges and

||f n+1 -f n || 2 2 = X |f n+1 -f n | 2 dµ = X c τn+1(x) (n + 1)λ -rn+1(x)+ρ τ n+1 (x) (n+1) -c τn(x) (n)λ -rn(x)+ρ τn (x) (n) 2 dµ = X c τn(x) (n) • c τn+1(x) (n + 1) c τn(x) (n) -λ rn+1(x)-rn(x)-ρ τ n+1 (x) +ρ τn (x) 2 dµ = d k=1 d l=1 h l (n)µ k (n + 1) s∈Sn(l,k) c l (n) • c k (n + 1) c l (n) -λ s,H(n) -ρ k (n+1)+ρ l (n) 2
Consequently, from the convergence of c l (n) to 1 as n → ∞ for l ∈ I and (5.2) one deduces,

n≥1 (l,k)∈J 1 M k,l (n + 1) s∈Sn(l,k) c k (n + 1) c l (n) -λ s,H(n) -ρ k (n+1)+ρ l (n) 2 converges. But | c k (n+1) c l (n) -λ s,H(n) -ρ k (n+1)+ρ l (n) | ≥ | c k (n+1) c l (n) -1|, then one also gets that n≥1 (l,k)∈J c k (n + 1) c l (n) -1 2 converges. One concludes that n≥1 (l,k)∈J 1 M k,l (n + 1) s∈Sn(l,k) |1 -λ s,H(n) -ρ k (n+1)+ρ l (n) | 2 .
converges, which gives the result.

Remark from last theorem that

1 M k,l (n+1)
s∈Sn(l,k) λ s,H(n) -ρ k (n+1)+ρ l (n) converges to 1 as n → ∞ for any (l, k) ∈ J. This suggests a strong condition on the distribution of powers of λ in S 1 in relation to the local ordering of the Bratteli-Vershik representation.

Example 1: measurable eigenvalues do not always come from the stable space

In Section 3 we proved that if λ = exp(2iπα) is a continuous eigenvalue of a minimal Cantor system (X, T ) given by a Bratteli-Vershik representation B = (V, E, ), then for some m ≥ 1 there exist v ∈ R C(m) with P (n, m)v → 0 as n → ∞ and w ∈ Z C(m) such that αH(m) = v + w. In this section we construct a uniquely ergodic Cantor minimal system of finite rank 2 and a measurable eigenvalue λ for which this property is not verified. In particular λ will not be a continuous eigenvalue.

We start by constructing a suitable sequence of matrices (M (n); n ≥ 1). Let

A = 1 1 1 0 . Let ϕ = 1+ √ 5 2
be the Perron eigenvalue of A, e u an associated eigenvector with positive coordinates and e s an eigenvector of the other eigenvalue ϕ -1 such that e s , (0, 1) T > 0 .

As usual H(1) = (1 1) T and for n ≥ 2 the matrix M (n) = A kn for some integer k n ≥ 2 to be defined. Recall P (1) = I and

P (n) = M (n) • • • M (2) for n ≥ 2 and P (n, m) = M (n) • • • M (m + 1) for 1 ≤ m ≤ n. We set K n = n i=2 k i , so P (n) = A Kn . For convenience we set k 1 = K 1 = 0.
In R 2 we distinguish the stable subspace E s = {v ∈ R 2 ; A n v → n→∞ 0} and the unstable subspace E u = {v ∈ R 2 ; ||A n v|| → n→∞ ∞} vector spaces of A. The vectors e u and e s belong respectively to the unstable and the stable spaces. Moreover, {e u , e s } is an orthonormal basis of R 2 .

Lemma 19. Let (ǫ n ; n ≥ 1) and (δ n ; n ≥ 1) be sequences of real numbers in ]0, ϕ -1 ] and v 1 ∈]0, ǫ 1 [. There exist a real number 0 < β < 1 and a sequence (k n ; n ≥ 1) of integers larger than 2 such that for all v > v 1 and all n ≥ 1

A Kn (βe u + ve s ) = z n + u n e u + (v n + ϕ -Kn (v -v 1 ))e s with 0 < v n < ǫ n , 0 < u n ≤ δ n v n and z n ∈ Z 2 .
Proof. First we construct recursively the sequence (k n ; n ≥ 1) and a sequence (α n ; n ≥ 1) such that for all n ≥ 1

z n = A Kn (α n e u + v 1 e s ) -v n e s ∈ Z 2
for some 0 < v n < ǫ n . We start the recursion with α 1 = 0, v 1 > 0 and z 1 = 0 and put t 1 = 1. Assume the construction is achieved up to n ≥ 1. Let k min ≥ 2 be large enough so that ϕ -kmin v n < ǫ n+1 and ϕ -kmin < ǫ n+1 . The direction given by e u has irrational slope. Thus there exist t n+1 > 0, 0 < s n+1 < ǫ n+1 -ϕ -kmin v n and zn+1 ∈ Z 2 such that t n+1 e u = zn+1 + s n+1 e s . Choose k n+1 > k min so that ϕ -kn+1 t n+1 < min( ϕ-1 ϕ δ n v n , ϕ -1 t n ), where t n is associated to z n in previous step. Let

v n+1 = ϕ -kn+1 v n + s n+1 and α n+1 = α n + ϕ -Kn+1 t n+1 .
One has

A Kn+1 (α n+1 e u + v 1 e s ) -v n+1 e s = (ϕ Kn+1 α n + t n+1 )e u + (ϕ -Kn+1 v 1 -ϕ -kn+1 v n -s n+1 )e s = ϕ kn+1 (ϕ Kn α n )e u + ϕ -kn+1 (ϕ -Kn v 1 -v n )e s + (t n+1 e u -s n+1 e s ) = A kn+1 [ϕ Kn α n e u + (ϕ -Kn v 1 -v n )e s ] + (t n+1 e u -s n+1 e s ) = A kn+1 z n + zn+1 = z n+1 ∈ Z 2 , and 0 < v n+1 = ϕ -kn+1 v n + s n+1 < ǫ n+1 . Let m ≥ 0. It holds, α n+m -α n = n+m-1 j=n (α j+1 -α j ) = n+m-1 j=n ϕ -Kj+1 t j+1 < ϕ -Kn+1 t n+1 m-1 j=0 ϕ -j < ϕ -Kn δ n v n < ϕ -Kn δ n ǫ n .
Then the sequence (α n ; n ≥ 1) converges. Put β = lim n→∞ α n . It is clear that 0 < β < 1 and ϕ Kn (β -α n ) < δ n v n for n ≥ 1. To conclude define u n = ϕ Kn (β -α n ) and observe that for all v > v 1

A Kn (βe u + ve s ) = z n + ϕ Kn (β -α n )e u + (v n + ϕ -Kn (v -v 1 ))e s
where by construction v n < ǫ n .

In previous lemma we gave a procedure to construct one value of β. In fact it is possible to construct a whole Cantor set of such numbers associated to a same sequence (k n ) n≥1 . The construction in the lemma can be modified as follows: at each step one finds two different values t n+1 > 0 and t ′ n+1 > 0 with t ′ n+1 > t n+1 and then we choose k n+1 large enough so that conditions for both values are satisfied. Remark that these conditions depend on all previous choices for t 2 , t ′ 2 , . . . , t n , t ′ n . Then we can set α n+1 = α n + ϕ -Kn+1 t n+1 as well as α ′ n+1 = α n + ϕ -Kn+1 t ′ n+1 . Since this choice is free at each step of the recurrence we can construct a Cantor set of values for β. It is straightforward that all the values obtained by this procedure are different. This argument shows that not all the possible values of β comes from the so called regular weak stable space, that is βH(1) / ∈ Z 2 + E s , because the intersection of Z 2 + E s with E u is countable. One has proved, Proposition 20. Let (ǫ n ; n ≥ 1) and (δ n ; n ≥ 1) be sequences of real numbers in ]0, ϕ -1 ] and 0 < v 1 < ǫ 1 . There exist a real number 0 < β < 1 such that βH(1) / ∈ Z 2 + E s and a sequence (k n ; n ≥ 2) of integers larger than 2 such that for all v > v 1 and all n ≥ 2

A Kn (βe u + ve s ) = z n + u n e u + (v n + ϕ -Kn (v -v 1 ))e s with 0 < v n < ǫ n , 0 < u n ≤ δ n v n and z n ∈ Z 2 .
Corollary 21. Let (ǫ n ; n ≥ 1) and (δ n ; n ≥ 1) be sequences of real numbers in ]0, ϕ -1 ]. There is a sequence (k n ; n ≥ 2) of integers larger than 2 and a real number α > 0 such that for all n ≥ 2 1) -e u || . Let 0 < β < 1 be given by Proposition 20 with v 1 and the sequences of epsilon's and delta's given there. We consider the intersection of {tH(1); t ∈ R} with {z + βe u + te s ; t ∈ R} where z = (1 0) T . Call it αH(1) = z + βe u + ve s . By construction one has v > v 1 . Then by Proposition 20 for n ≥ 2, αP (n)H(1

αP (n)H(1) = w n + z n , where z n ∈ Z 2 and w n ∈ R 2 with ||w n || ≤ 4ǫ n . Proof. Let v 1 < min ǫ 1 , || 1 eu,H(1) H(
) = P (n)z + z n + u n e u + (v n + ϕ -Kn (v -v 1 ))e s . Thus as ϕ -Kn ≤ ǫ n , v n ≤ ǫ n and u n ≤ δ n ǫ n one concludes.
Now the matrices (M (n); n ≥ 2) have been constructed we will proceed to give an ordering to the Bratteli diagram induced by them. We introduce the notion of best ordering associated to (w, h) where w = (w 1 , w 2 ) T ∈ R 2 with w 2 ≥ 0 and h = (h 1 , h 2 ) T ∈ N 2 with strictly positive coordinates such that the slope f = |w 1 |/|w 2 | of {w} ⊥ is smaller than h 2 /h 1 . This ordering is described by a word p = p 1 . . . p l 1 in {1, 2} * of length l = h 1 + h 2 defined recursively by: set p 0 = 0 and for 0 ≤ n ≤ l -1

p n+1 = 1 if ( n i=1 e pi -h), w > 0 2 otherwise
where e 1 , e 2 are the canonical vectors of R 2 . Let w ⊥ be a vector orthogonal to w. Consider the line L = {h + tw ⊥ ; t ∈ R}. Notice that, since w 2 ≥ 0, a point y ∈ R 2 is above this line if and only if y -h, w > 0. Thus p n+1 = 1 if the integer vector n i=1 e pi is above the line L and is equal to 2 otherwise. In particular, since h, w > 0 then p 1 = 2. This motivates the following definition:

K(p) = inf {i ≥ 1 : p i = 1} -2. Lemma 22. It holds, • K(p) ≤ h 2 + sign(w 1 )f h 1 ≤ h 1 ( h2 h1 + sign(w 1 )f ); • for all j ≥ K(p), l i=j e pi , w ≤ ||w||.
Proof. The intersection point of L with {(0, 1) T } is h,w w2 (0, 1) T . This gives the first inequality. The second one follows directly when computing the orthogonal projection of l i=j e pi over {tw; t ∈ R}. Fix two decreasing sequences of real numbers (ǫ n ; n ≥ 1) and (δ n ; n ≥ 1) with values in ]0, ϕ -1 ] such that δ n ≤ ǫ n for n ≥ 1. Let 0 < v 1 < ǫ 1 . Let α and (k n ; n ≥ 1) be as in Corollary 21. Then αP (n)H(1) = w n + z n where z n ∈ Z 2 and w n ∈ R 2 for n ≥ 2. From construction it follows that (w n ) 2 > 0. Let (X, T ) be the minimal Cantor system defined from the ordered Bratteli-Vershik diagram described as follows: (i) the vertex at each level are labelled by {1, 2}, (ii) the incidence matrices are given by M (n) = A kn for n ≥ 2, and (iii) for j ∈ {1, 2} and n ≥ 2 the order of the m j (n) = (M j,1 (n), M j,2 (n)) T edges arriving at vertex j in level n is given by the best order associated to (w n , m n (j)) described by the word p (n,j) . Since p (n,j) 1 = 2 and p (n,j) l+1 = 1 this diagram has unique minimal and maximal points. Thus for any x ∈ X and n ≥ 1 its suffix s n (x) is given by,

s n (x) = mj (n),H(1) k=on(x)+1 e p (n,j) k + e 1 ,
where τ n+1 (x) = j and o n (x) is the order of x n+1 . Thus, given i, j ∈ {1, 2}, all vectors of type γ n = mj (n),H(1) k=o e p (n,j) k with 1 ≤ l ≤ m j (n), H(1) are the suffix s n (x) of some x ∈ X with τ n (x) = i and τ n+1 (x) = j where i is such that p (n,j) o = i. Let µ be the unique invariant measure of (X, T ) (it is unique since µ(n), e s = 0 for all n ≥ 1). A direct computation yields to

µ{s n = γ n | τ n = i, τ n+1 = j} = 1 M j,i (n + 1) .
Lemma 23. There is a positive constant C such that for all n ≥ 1 ,j) ). From the second statement of Lemma 22 one gets

µ{ s n , w n > ||w n ||} ≤ Cǫ n . Proof. Let i, j ∈ {1, 2}. Set K j (n) = K(p (n
µ{ s n , w n > ||w n || | τ n = i, τ n+1 = j} ≤ µ{1 ≤ o n < K j (n) | τ n = i, τ n+1 = j} ≤ |{1 ≤ o < K j (n) ; p (n,j) o = i}| M j,i (n + 1) If τ n (x) = 1 then necessarily o n (x) > K j (n), while if τ n (x) = 2 then |{1 ≤ o < K j (n) ; p (n,j) o = 2}| = K j (n) -1. So in this case µ{ s n , w n > ||w n || | τ n = i, τ n+1 = j} ≤ K j (n) M j,2 (n + 1)
.

Let f n be the slope of the orthogonal line defined from w n . By construction (it is not difficult to verify) one has (w n ) 1 < 0. Then from Lemma 22 one gets

K j (n) M j,2 (n + 1) ≤ M j,1 (n + 1) M j,2 (n + 1) M j,2 (n + 1) M j,1 (n + 1) -f n ≤ M j,1 (n + 1) M j,2 (n + 1) M j,2 (n + 1) M j,1 (n + 1) -ϕ -1 + ϕ -1 -f n .
Let w n = vn e s + u n e u . Recall from construction that vn =

v n + ϕ -Kn (v -v 1 ), v n ≤ ǫ n and u n ≤ δ n v n . Also ϕ -kn ≤ ǫ n .
The slope f n is given by

f n = ϕ -1 vn -u n vn + ϕ -1 u n = ϕ -1 -un vn 1 + ϕ -1 un vn . Thus |f n -ϕ -1 | = u n vn 1 + ϕ -2 1 + ϕ -1 un vn ≤ u n v n (1 + ϕ -2 ) ≤ (1 + ϕ -2 )δ n ≤ (1 + ϕ -2 )ǫ n .
On the other hand,

Mj,2(n+1) Mj,1(n+1) approaches ϕ -1 at speed ϕ -kn+1 ≤ ǫ n+1 ≤ ǫ n . Thus K j (n) M j,2 (n + 1) ≤ Cǫ n where C = 2 + ϕ -2 .
To conclude, one integrates this uniform bound with respect to i and j.

Remark 24. In general, the quantities s n (x), w n are not bounded. But it is more likely that a point taken at random has s n (x), w n of order ||w n ||.

One assumes (ǫ n ; n ≥ 1) is summable ( n≥1 ǫ n < ∞).

Theorem 25. The complex number exp(2iπα) is an eigenvalue of (X, T, µ) that is not continuous.

Proof. The fact that it is not continuous follows directly from construction and Theorem 5. Hence the series converges almost surely.

To conclude recall f (x) = exp(-2iπ n≥1 s n (x), αP (n)H( 1) ) is an eigenfunction of (X, T ) associated to exp(2iπα).

7. Example 2: continuous and measurable eigenvalues of Toeplitz type systems of finite rank

It is known that any subgroup of S 1 can be the set of measurable eigenvalues of a Toeplitz system (see [DL] or [Dow]). The main motivation of this section is to show a class of examples of Toeplitz Cantor minimal systems where the finite rank assumption restricts the possibilities of measurable eigenvalues.

An ordered Bratteli-Vershik diagram is of Toeplitz type if for all n ≥ 1 and for all u, v ∈ V n the number of edges in E n-1 finishing at u coincides with the number of edges in E n-1 finishing at v. Denote this number q n and set p n = q n q n-1 • • • q 1 . We say (q n ; n ≥ 1) is the characteristic sequence of the diagram. A Cantor minimal system is said to be of Toeplitz type if it is given by a Bratteli-Vershik diagram of this type. This definition is motivated by the characterization of Toeplitz subshifts in [GJ]. That is, a Bratteli-Vershik diagram of Toeplitz type is a Toeplitz subshift whenever it is expansive. First we prove a known result for Toeplitz subshifts.

Theorem 26. Let (X, T ) be a Cantor minimal system of Toeplitz type given by a Bratteli-Vershik system with characteristic sequence (q n ; n ≥ 1). Then, exp(2iπα) is a continuous eigenvalue of (X, T ) if and only if α = a pn for some a ∈ Z and n ≥ 1.

Proof. Let exp(2iπα) be a continuous eigenvalue of (X, T ) with α ∈]0, 1[. Let α = i≥1 ai pi with a i ∈ {0, . . . , q i -1} for all i ≥ 1 be the expansion of α in base (p n ; n ≥ 1). By Theorem 5 one has that αp n → 0 mod Z as n → ∞. This implies that i≥n+1 ai qn+1•••pi → n→∞ 0. Recall H(1) = (1, . . . , 1) T . From Proposition 1 one knows that αp n s n (x), H(1) converges to 0 modulo Z and uniformly in x. Let x n ∈ X such that s n (x n ), H(1) = b n+1 = ⌊ qn+1 2an+1 ⌋. It exists since s n (x), H(1) can take any value between {0, . . . , q n+1 -1}. If (a n ; n ≥ 1) is not ultimately equal to 0, then lim n→∞ αp n s n (x n ), H(1) = 1 2 , that contradicts the fact that it is 0 modulo Z. One concludes (a n ; n ≥ 1) is ultimately equal to 0 and that α = a pm for some a ∈ N and m ∈ N. Conversely, assume α = a pm for some a ∈ N and m ∈ N. Then for all x ∈ X and n ≥ m one has

s n (x), αP (n)H(1) = a p n p m s n (x), H(1) = aq m+1 • • • q n s n (x), H(1) ,
which belongs to Z. Then n≥1 s n (x), αP (n)H(1) converges uniformly modulo Z. One concludes by using Proposition 1.

Let (X, T ) be a minimal Cantor system given by a Bratteli-Vershik diagram of Toeplitz type. The next proposition shows that in the class of linearly recurrent systems of Toeplitz type, continuous and measurable eigenvalues coincide.

Theorem 27. Let (X, T ) be a Toeplitz type system with finite rank and µ be the unique T -invariant probability measure. Let (q n ; n ≥ 1) be the characteristic sequence of the associated diagram and suppose it is bounded. Then exp(2iπα) is an eigenvalue of (X, T, µ) if and only if α = a pm for some a ∈ Z and m ∈ N. In particular, they are all continuous eigenvalues. Proof. Let exp(2iπα) be a measurable eigenvalue with α ∈]0, 1[. and α = i≥1 ai pi with a i ∈ {0, . . . , q i -1} for all i ≥ 1 be the expansion of α in base (p n ; n ≥ 1). From [BDM] one knows that αP (n)H( 1), e 1 = p n α → n→∞ 0 mod Z. This implies that an qn goes to zero with n goes to infinity. The characteristic sequence being bounded one concludes (a n ; n ≥ 1) is ultimately equal to 0 and that α = a pm for some a ∈ N and m ∈ N. We conclude using Theorem 26.

Let (X, T ) be a minimal Cantor system of Toeplitz type of finite rank d and let µ be a T -ergodic probability measure. Let (q n ; n ≥ 1) be the characteristic sequence of the associated Bratteli-Vershik diagram. Consider λ = exp(2iπα) to be a measurable eigenvalue of (X, T, µ) and f : X → C to be an associated eigenfunction with |f | = 1, µ-almost surely. One has that f n = E µ (f |T n ) converges µ-almost surely and in L 2 (X, B(X), µ) to f . Following the notations of Section 5, we recall

f n (x) = B k (n) f dµ µ k (n) λ -j = c k (n)λ ρ k (n)-j whenever x ∈ T -j B k (n) for some 1 ≤ k ≤ d and 0 ≤ j < h k (n). We set c ′ k (n) = c k (n)λ ρ k (n) . Remark j = n-1 i=1 s i (x), P (i)H(1) = n-1 i=1 p i s i (x), H(1) = n-1 i=1 p i si (x),
where si (x) = s i (x), H(1) . Since the system is of Toeplitz type one knows that 0 ≤ si (x) < q i+1 . Given 1 ≤ i, k ≤ d and n ≥ 2 define S k,i (n) = {s n (x) : x ∈ X, τ n (x) = k, τ n+1 (x) = i}. Let I = {i ∈ {1, . . . , d}; lim inf n→∞ µ{τ n = i} > 0}. Contracting the Bratteli-Vershik diagram given (X, T ) if needed we can assume n≥1 µ{τ n = i} < ∞ for all i ∈ I c , that is, the representation of (X, T ) can be assume clean.

Theorem 28. Let (X, T ) be a Cantor minimal system of Toeplitz type of finite rank d and let µ be a T -ergodic probability measure. Then all measurable eigenvalues of (X, T, µ) are rational.

Proof. Contracting if necessary we can assume (X, T ) is represented by a clean Bratteli-Vershik diagram of Toeplitz type. Let (q n ; n ≥ 1) be the characteristic sequence of the diagram and λ = exp(2iπα) be a measurable eigenvalue of (X, T, µ). From martingale theorem f n → f as n → ∞ µ-a.e and since the Bratteli-

Vershik diagram given (X, T ) is clean then µ{x ∈ X; lim n→∞ |c ′ τn(x) (n)| = 1} = 1. Hence, by Egoroff theorem, for ρ < 1 8d 2 there is a measurable set A such that µ(A) > 1 -ρ and (f n ) converges to f and |c ′ τn(x) (n)| converges to 1 as n → ∞ uniformly on A. Let ǫ = 1 8d 4 .
Then for all n < N large enough and x ∈ A one has |f n (x)-f N (x)| ≤ ǫ and |c ′ τn(x) (n)| > 2/3. By using the expression of f n we recall before, one has

c ′ τN (x) (N ) c ′ τn(x) (n) -(λ pn ) sn,N (x) ≤ ǫ |c ′ τn(x) (n)| for every x ∈ A, where sn,N (x) = N -1 i=n q n+1 • • • q i si (x). Put Q n,N = q n+1 • • • q N . Clearly 0 ≤ sn,N < Q n,N .
Assume α is irrational. For any interval L ⊆ S 1 , by the unique ergodicity of the rotation by λ pn , one has

d n,N (L) = 1 Q n,N |{0 ≤ s ≤ Q n,N -1 : λ pns ∈ L}| → N →∞ |L|
uniformly in L of the same length.

Let L be an interval in S 1 such that |L| = 1 4d 2 and fix N such that d n,N (L) > |L|/2. In addition we can assume assume the interval L is disjoint from the set

c ′ i(N )
c ′ j (n) : 1 ≤ i, j ≤ d and that the distance between L and ci(N ) cj(n) : 1 ≤ i, j ≤ d is bigger than 2ǫ. Therefore, µ{x ∈ X; λ pn sn,N (x) ∈ L} = This implies that µ{x ∈ A; λ pn sn,N (x) ∈ L} > 0 and thus there is x ∈ A such that λ pn sn,N (x) ∈ L while

2ǫ ≤ c ′ τN (x) (N ) c ′ τn(x) (n) -(λ pn ) sn,N (x) ≤ ǫ |c ′ τn(x) (n)| .
Thus, |c ′ τn(x) (n)| ≤ 1/2, which is a contradiction. One concludes α is rational. Let us now show that the measurable eigenvalues can not be any rational numbers.

Proposition 29. Let (X, T ) be a Cantor minimal system of Toeplitz type of finite rank d and µ a T -ergodic probability measure. Let (q n ; n ≥ 1) be the characteristic sequence of the associated diagram. If exp(2iπ(p/q)), with (p, q) = 1, is a non continuous rational eigenvalue of (X, T, µ) then for all n large enough, q (q,pn) ≤ d. Proof. Let λ = exp(2iπp/q), p, q ∈ N, (p, q) = 1, be a non continuous eigenvalue of (X, T, µ). Thus exp(2iπ/q) is a non continuous eigenvalue of (X, T, µ) and we can suppose p = 1. From Theorem 26 we deduce exp(2iπ(q, p n )/q) is a non continuous eigenvalue for all n large enough. Hence we can assume (q, p n ) = 1 for all n large enough. Fix n ≥ 1. Let ǫ > 0 be such that ǫ < 1/2qd. Contracting the diagram if needed, one can suppose q/q n < ǫ for all n ≥ 1. Recall r n+1 (x) -r n (x) = sn (x)p n with 0 ≤ sn ≤ q n+1 -1. For all 0 ≤ a ≤ q -1 set S n (a) = {x ∈ X; sn = a mod q}. Let q n+1 = kq + r with 0 ≤ r ≤ q -1. For all t ∈ {1, . . . , d} one has that

1 q -ǫ ≤ 1 q - 1 q n+1 ≤ 1 q - r qq n+1 ≤ k q n+1
≤ µ{S n (a)|τ n+1 = t}. (7.1) From Theorem 7 in [BDM], there exist real functions ρ n : {1, . . . , d} → R such that ((1/q)r n -ρ n ; n ∈ N) converges µ-almost everywhere modulo Z. Thus, from Egoroff theorem there exists A ∈ B(X) with µ(A) ≥ 1 -ǫ such that ((1/q)r n -ρ n ; n ∈ N) converges uniformly on A in R/Z. Thus ((1/q)s n (x)-(ρ n+1 -ρ n ); n ∈ N) converges uniformly to 0 on A in R/Z. There exists t ∈ {1, . . . , d} such that µ{τ n+1 = t} ≥ 1/d. Hence, using (7.1), one obtains µ(S n (a) ∩ {τ n+1 = t} ∩ A) ≥ ( 1 q -ǫ)/d -ǫ ≥ 1 qd -2ǫ > 0 for any 0 ≤ a ≤ q -1. Suppose q > d. Then there exist x ∈ S n (a)∩{τ n+1 = t}∩A and y ∈ S n (b)∩{τ n+1 = t} ∩ A, with a = b, and τ n (x) = τ n (y). Then, pn q (a -b) should go to 0 in R/Z because p n q (a -b) = p n q ( sn (x) -sn (y)) -(ρ n+1 (x) -ρ n (x)) + (ρ n+1 (y) -ρ n (y)) → 0 in R/Z. But this is not possible because 1 ≤ |a -b| ≤ q -1 and (q, p n ) = 1. Hence q ≤ d.

The last theorem implies that an arbitrary subgroup of S 1 cannot be the set of eigenvalues of a Toeplitz minimal system of finite rank. Also, it is not difficult to deduce from last theorem that there is a unique q ≤ d with (q, p n ) = 1 for all enough large n such that all non continuous eigenvalues of the same type are in the subgroup generated by 1/q. Finally observe that from [DM] it follows that Toeplitz type Cantor minimal systems of finite rank that have non continuous eigenvalues are expansive (thus subshifts).

In the following example we provide a Toeplitz system of finite rank 3 where λ = -1 is a non continuous eigenvalue.

Example. Let (l n ; n ≥ 1) be a strictly increasing sequence of integers with l 1 = 0. Put q n = 3 ln for n ≥ 1. Consider the Toeplitz system (X, T, µ) of finite rank 3 given by the Bratteli-Vershik diagram with characteristic sequence (q n ; n ≥ 1) and such that each tower of level n is built by concatenating towers of previous level in the following way:

1 → (12) tn-3 131, 2 → 1(12) tn-3 31, 3 → (12) tn-3 131, where q n = 2t n -3. We set Q n = q 1 q 2 • • • q n for all n. Let n ≥ 1. Define ρ 1 (n) = -ρ 2 (n) = -ρ 3 (n) = 1 and f n (x) = (-1) j ρ k (n) if x ∈ T -j B k (n) for k ∈ {1, 2, 3} and 0 ≤ j < h k (n). We set A n = {x ∈ X; f n (x) = f n+1 (x)} = ∪ 1≤i,j≤3 A n (i, j) where

A n (i, j) = {x ∈ X; τ n (x) = i, τ n+1 (x) = j, f n (x) = f n+1 (x)}.

Let x ∈ A n (1, 1). We have f n (x) = (-1) j ρ 1 (n) and f n+1 (x) = (-1) j+2l3 n ρ 1 (n + 1) = f n (x) for some l ∈ N. Studying the other cases one can check that

A n = {τ n+1 = 3} ∪ {τ n+1 = 2, τ n = 3} ∪    0≤k≤Qn -1 Q n+1 -Qn ≤k≤Q n+1 -1 T -k B 2 (n + 1)    and µ(A n ) = 1 q n+2 + 1 q n+1 µ(τ n+1 = 2) + 2 q n+1 µ(τ n+1 = 2) ≤ 4 q n+1 .
As 1 qn+1 converges one deduces that µ(lim sup n→∞ A n ) = 0. Then, (f n ) converges µ-almost everywhere and one can check that f • T = -f µ-almost everywhere. Hence, -1 is a measurable eigenvalue of the system. Theorem 26 implies -1 is not a continuous eigenvalue.

  First we prove the seriesn≥1 ||| s n (x), αP (n)H(1) ||| converges µ-almost surely. Since n≥1 µ{ s n , w n > ||w n ||} ≤ n≥1 ǫ n < ∞, then by Borel-Cantelli Lemma one has for µ-almost x ∈ X n≥1 1 { sn(x),wn >||wn||} < ∞.Denote by N 0 (x) the first integer such that for all N > N 0 (x), s n (x), w n ≤ ||w n ||. Since N 0 is almost surely finite, one has, for µ-almost all x ∈ X and N > N 0 (x),n>N ||| s n (x), αP (n)H(1) ||| ≤ n>N | s n (x), w n | ≤ n>N ||w n || ≤ n>N ǫ n .

  N )p n Q n,N d n,N (L) = d n,N (L) > |L| 2 > ρ .

Acknowledgments. The third author is supported by Nucleus Millennium Information and Randomness P04-069-F. This project was also partially supported by the international cooperation program ECOS-Conicyt C03-E03.