We investigate weak mixing for some classes of interval translation mappings. We give two distinct proofs that a typical Bruin-Troubetzkoy interval translation mapping is weakly mixing. Moreover, we show that the second approach extends to other classes of interval translation mappings. In particular, we show that Bruin interval translation mappings on any number of intervals are typically weak mixing. Finally, we construct the first examples of non weak mixing Bruin-Troubetzkoy ITM of infinite type.

On the full shift on two symbols, we consider the potential defined by $V(x) = \frac{1}{n}$ where $n$ denotes the longest common prefix between the infinite word $x$ and an element of the subshift associated to the Thue-Morse substitution. Given a non negative real number $β$, the pressure function is $P(β):=\sup\left\{h_μ+β\int V\,dμ\right\},$ where the supremum is taken over all shift invariant probabilities $μ$ on the full shift and $h_μ$ is the Kolmogorov entropy. We prove that there is a freezing phase transition for the potential $V$: For $β$ large enough, the pressure $P(\be)$ is equal to zero. Similar results were previously published by Bruin and Leplaideur in \cite{BL2}, \cite{Bruin-Leplaid-13} but their proofs contained significant gaps and required substantial clarification.

We compute the complexity of the billiard language of the regular Euclidean N-gons (and other families of rational lattice polygons), answering a question posed by Cassaigne-Hubert-Troubetzkoy. Our key technical result is a counting result for saddle connections on lattice surfaces, when we count by combinatorial length.

We show that any real number in [0,1) is a diffusion rate for the wind-tree model with rational parameters. We will also provide a criterion in order to describe the shape of the Lyapunov spectrum of cocycles obtained as suspension of a representation. As an application, we exhibit an infinite family of wind-tree billiards for which the interior of the Lyapunov spectrum is a big as possible: this is the full square (0,1)^2. To the best of the knowledge of the authors, these are the first complete descriptions where the interior of the Lyapunov spectrum is known explicitly in dimension two, even for general Fuchsian groups.

We study a class of interval translation mappings introduced by Bruin and Troubetzkoy, describing a new renormalization scheme, inspired by the classical Rauzy induction for this class. We construct a measure, invariant under the renormalization, supported on the parameters yielding infinite type interval translation mappings in this class. With respect to this measure, a.e. transformation is uniquely ergodic. We show that this set has Hausdorff dimension between 1.5 and 2, and that the Hausdorff dimension coincides with the affinity dimension. Finally, seeing our renormalization as a multidimensional continued fraction algorithm, we show that it has almost always the Pisot property. We discover an interesting phenomenon: the dynamics of this class of transformations is often (conjecturally: almost always) weak mixing, while the renormalizing algorithm typically has the Pisot property.

This book explores infinite-type translation surfaces and is intended as an introductory text for graduate and PhD students, as well as a reference for more advanced researchers. Chapter 1 introduces the three definitions of translation surfaces and meticulously proves their equivalence. It is enriched with numerous examples that are revisited throughout the book. Chapter 2 provides a detailed examination of the topological classification of infinite-type surfaces, the construction of infinite coverings of finite-type translation surfaces, and the structure of points within the metric completion. Chapter 3 investigates the affine symmetries of infinite-type translation surfaces, with special emphasis on infinite coverings of finite-type surfaces, the Hooper-Thurston-Veech construction, and affine homeomorphisms of finite-area infinite-type translation surfaces. Chapter 4 introduces infinite interval exchange transformations and employs them to demonstrate that the dynamics of translation flows are significantly more complex in the infinite-type context. The two appendices address hyperbolic geometry and the spectra of infinite graphs, respectively.

We give conditions for minimality of ℤ/Nℤ extensions of a rotation of angle α with one marked point, solving the problem for any prime N: for N=2 , these correspond to the Veech 1969 examples, for which a necessary and sufficient condition was not known yet. We provide also a word combinatorial criterion of minimality valid for general interval exchange transformations, which applies to ℤ/Nℤ extensions of any interval exchange transformation with any number of marked points. Then we give a condition for unique ergodicity of these extensions when the initial interval exchange transformation is linearly recurrent and there are one or two marked points.

Let $\Omega$ be a strictly convex divisible subset of the $n$-dimensional real projective space which is not an ellipsoid. Even though $\partial\Omega$ is not $C^2$, Benoist showed that it is $C^{1+\alpha}$ for some $\alpha>0$, and Crampon established that $\partial\Omega$ actually possesses a sort of anisotropic H\"older regularity -- described by a list $\alpha_1\leq\dots\leq\alpha_{n-1}$ of positive real numbers -- at almost all of its points. In this article, we show that $\partial\Omega$ is maximally anisotropic in the sense that this list of approximate regularities of $\partial\Omega$ does not contain repetitions. This result is a consequence of the simplicity of the Lyapunov spectrum of the Hilbert geodesic flow for every equilibrium measure associated to a H\"older potential.

We define a condition on the resolution of bispecials in a language. A language satisfies this order condition if and only if it is the natural coding of a generalized interval exchange transformation, while the order condition plus some additional ones characterize the codings of various more classical interval exchange transformations. Also, a finite word clusters for the Burrows-Wheeler transform if and only if the language generated by its powers satisfies an order condition.

At the beginning of the 80s, H. Masur and W. Veech started the study of generic properties of interval exchange transformations (IETs) proving that almost every such transformation is uniquely ergodic. About the same time, S. Novikov's school and French mathematicians independently discovered very intriguing phenomena for classes of measured foliations on surfaces and respective IETs. For instance, minimality is exceptional in these families. A precise version of this statement is a conjecture by Novikov. The French and Russian constructions are very different ones. Nevertheless, in the most simple situation (surfaces of genus three with two singularities) it was recently observed that both foliations share the same type of properties. For instance, the space of minimal parameters is the same, called the Rauzy gasket. However, the precise connection between these two series of works was rather unclear. The aim of this paper is to prove that both theories describe, in different languages, the same objects. This text provides an explicit dictionary between both constructions.

The Arnoux-Rauzy systems are defined in [6], both as symbolic systems on three letters and exchange transformations of six intervals on the circle. In connection with a conjecture of S.P. Novikov, we investigate the dynamical properties of these interval exchange transformations, and precise their relation with the symbolic systems, which was known only to be a semi-conjugacy. In order to do this, we define a new system which is an exchange transformation of nine intervals on the line (it was described in [4] for a particular case). Our main result is that the semi-conjugacy determines a measure-theoretic isomorphism (between the three systems) under a diophantine (sufficient) condition, which is satisfied by almost all Arnoux-Rauzy systems for a suitable measure. However, under another condition, the interval exchange transformations are not uniquely ergodic and the isomorphism does not hold for all invariant measures. Finally, we give conditions for these interval exchange transformations to be weakly mixing.

We introduce a new renormalization procedure on double rotations, which is reminiscent of the classical Rauzy induction. Using this renormalization we prove that the set of parameters which induce infinite type double rotations has Hausdorff dimension strictly smaller than 3 3 . Moreover, we construct a natural invariant measure supported on these parameters and show that, with respect to this measure, almost all double rotations are uniquely ergodic.

We look at d -point extensions of a rotation of angle α with r marked points, generalizing the examples of Veech 1969 and Sataev 1975, together with the square-tiled interval exchange transformations of [5]. We study the property of rigidity, as a function of the Ostrowski expansions of the marked points by α : we prove that T is rigid when α has unbounded partial quotients, and that T is not rigid when the natural coding of the underlying rotation with marked points is linearly recurrent. But there remains an interesting gray zone between these two cases, in which we have only partial results on the rigidity question; they allow us to build the first examples of non linearly recurrent and non rigid interval exchange transformations.

AbstractIn this we exploit the arithmeticity criterion of Oh and Benoist–Miquel to exhibit an origami in the principal stratum of the moduli space of translation surfaces of genus three whose Kontsevich–Zorich monodromy is not thin in the sense of Sarnak.

In this paper, we give a geometric criterion ensuring the recurrence of the vertical flow on Zd-covers of compact translation surfaces (d≥2). We prove that the linear flow in the wind-tree model is recurrent for every pair of parameters and almost every direction.

The Arnoux-Rauzy systems are defined in \cite{ar}, both as symbolic systems on three letters and exchanges of six intervals on the circle. In connection with a conjecture of S.P. Novikov, we investigate the dynamical properties of the interval exchanges, and precise their relation with the symbolic systems, which was known only to be a semi-conjugacy; in order to do this, we define a new system which is an exchange of nine intervals on the line (it was described in \cite{abb} for a particular case). Our main result is that the semi-conjugacy determines a measure-theoretic isomorphism (between the three systems) under a diophantine (sufficient) condition, which is satisfied by almost all Arnoux-Rauzy systems for a suitable measure; but, under another condition, the interval exchanges are not uniquely ergodic and the isomorphism does not hold for all invariant measures; finally, we give conditions for these interval exchanges to be weakly mixing.

We look at interval exchange transformations defined as first return maps on the set of diagonals of a flow of direction $\theta$ on a square-tiled surface: using a combinatorial approach, we show that, when the surface has at least one true singularity both the flow and the interval exchange are rigid if and only if tan $\theta$ has bounded partial quotients. Moreover, if all vertices of the squares are singularities of the flat metric, and tan $\theta$ has bounded partial quotients, the square-tiled interval exchange transformation T is not of rank one. Finally, for another class of surfaces, those defined by the unfolding of billiards in Veech triangles, we build an uncountable set of rigid directional flows and an uncountable set of rigid interval exchange transformations.

In this paper we prove a central limit theorem for some probability measures defined as asymptotic densities of integer sets defined via sum-of-digit-function. To any non-negative integer a we can associate a measure on Z called mu(a) such that, for any d, mu(a) (d) is the asymptotic density of the set of non-negative integers n such that s(2)(n + a) - s(2)(n) = d where s(2) (n) is the number of digits "1" in the binary expansion of n. We express this probability measure as a product of matrices whose coefficients are operators of l(1)(Z). Then we take a sequence of integers (a(X)(n))(n is an element of N) defined via a balanced Bernoulli sequence X. We prove that, for almost every sequence, and after renormalization by the typical variance, we have a central limit theorem by computing all the moments and proving that they converge towards the moments of the normal law N(0, 1).

Consider a periodic tiling of a plane by equal triangles obtained from the equilateral tiling by a linear transformation. We study a following tiling billiard: a ball follows straight segments and bounces of the boundaries of the tiles into neighbouring tiles in such a way that the coefficient of refraction is equal to -1. We show that almost all the trajectories of such a billiard are either closed or escape linearly, and for closed trajectories we prove that their periods belong to the set 4N+2. We also give a precise description of the exceptional family of trajectories (of zero measure) : these trajectories escape non-linearly to infinity and approach fractal-like sets. We show that this exceptional family is parametrized by the famous Rauzy gasket. This proves several conjectures stated previously on triangle tiling billiards. In this work, we also give a more precise understanding of fully flipped minimal exchange transformations on 3 and 4 intervals by proving that they belong to a special hypersurface. Our proofs are based on the study of Rauzy graphs for interval exchange transformations with flips.

In this paper we study correlation measures introduced in . Denote by μ_a(d) the asymptotic density of the set ℰ_a,d={n ∈ℕ, s_2(n+a)-s_2(n)=d} (where s_2 is the sum-of-digits function in base 2). Then, for any point X in {0,1}^ℕ, define the integer sequence (a_X (n))_n∈ℕ such that the binary decomposition of a_X (n) is the prefix of length n of X. We prove that for any shift-invariant ergodic probability measure ν on {0,1}^ℕ, the sequence (μ_a_X(n))_n ∈ℕ satisfies a central limit theorem. This result was proven in the case where ν is the symmetric Bernoulli measure in .