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Pascal Hubert

Publications and source records attributed to Pascal Hubert.

At least 19 recordsLinked to original sources
arXiv2026

Typical Weak Mixing and Exceptional Spectral Properties for Interval Translation Mappings

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.

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arXiv2025

Freezing phase transition for the Thue-Morse subshift

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 $\beta$, the pressure function is $P(\beta):=\sup\left\{h_{\mu}+\beta\int V\,d\mu\right\},$ where the supremum is taken over all shift invariant probabilities $\mu$ on the full shift and $h_{\mu}$ is the Kolmogorov entropy. We prove that there is a freezing phase transition for the potential $V$: For $\beta$ 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.

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arXiv2025

Linear Flows on Translation Prisms

Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat $3$-manifolds which we call translation prisms. Using ideas of Furstenberg and Veech, we connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. Furthermore, we construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in a right prism over a regular $n$-gons for $n=7, 9, 14, 16, 18, 20, 24, 30$.

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arXiv2025

Complexity for billiards in regular N-gons

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.

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arXiv2024

Renormalization for Bruin-Troubetzkoy ITMs

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.

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arXiv2024

Infinite Translation Surfaces in the Wild

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.

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arXiv2023

Diffusion rate in non-generic directions in the wind-tree model

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.

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arXiv2023

Simplicity of Lyapunov spectra and boundaries of non-conical strictly convex divisible sets

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.

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arXiv2022

Languages of general interval exchange transformations

The languages generated by interval exchange transformations have been characterized by Ferenczi-Zamboni (2008) and Belov-Cernyatev (2010) under some extra conditions on the system. Lifting these conditions leads us to consider successively natural codings of standard interval exchange transformations, natural codings of affine interval exchange transformations, grouped codings of affine interval exchange transformations, and natural codings of generalized interval exchange transformations. We show that these four classes of languages are strictly increasing, and give necessary and/or sufficient (but not all equally explicit) combinatorial criteria to describe each of them. These work also, mutatis mutandis, for interval exchanges with flips

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arXiv2021

Minimality and unique ergodicity of Veech 1969 type interval exchange transformations

We give conditions for minimality of $\mathbb Z/N\mathbb Z$ extensions of a rotation of angle $\alpha$ 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 $\mathbb Z/N\mathbb Z$ 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.

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arXiv2021

A note on double rotations of infinite type

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. 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.

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arXiv2020

Dynamical systems around the Rauzy gasket and their ergodic properties

At the beggining of the 80's, H.Masur and W.Veech started the study of generic properties of interval exchange transformations 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.

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arXiv2019

Arnoux-Rauzy interval exchange transformations

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.

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arXiv2018

Normal distribution of correlation measures of binary sum-of-digits functions

In this paper we study correlation measures introduced in \cite{emme_asymptotic_2017}. Denote by $\mu_a(d)$ the asymptotic density of the set $\mathcal{E}_{a,d}=\{n \in \mathbb{N}, \ s_2(n+a)-s_2(n)=d\}$ (where $s_2$ is the sum-of-digits function in base 2). Then, for any point $X$ in $\{0,1\}^\mathbb{N}$, define the integer sequence $\left(a_X (n)\right)_{n\in \mathbb{N}}$ such that the binary decomposition of $a_X (n) $ is the prefix of length $n$ of $X$. We prove that for \textit{any} shift-invariant ergodic probability measure $\nu$ on $\{0,1\}^\mathbb{N}$, the sequence $\left(\mu_{a_X(n)}\right)_{n \in \mathbb{N}}$ satisfies a central limit theorem. This result was proven in the case where $\nu$ is the symmetric Bernoulli measure in \cite{emme_central_2018}.

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arXiv2018

Triangle tiling billiards and the exceptional family of their escaping trajectories: circumcenters and Rauzy gasket

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.

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arXiv2018

Les exposants de Liapounoff du flot de Teichm\"{u}ller

The purpose of the paper under review is to explain the main ideas and the main ingredients of the involved and delicate work of A. Eskin, M. Kontsevich and A. Zorich concerning the sum of the positive Lyapunov exponents of the so-called Kontsevich- Zorich cocycle acting on the first cohomology spaces of translation surfaces.

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arXiv2017

Limit laws for random matrix products

In this short note, we study the behaviour of a product of matrices with a simultaneous renormalization. Namely, for any sequence $(A\_n)\_{n\in \mathbb{N}}$ of $d\times d$ complex matrices whose mean $A$ exists and whose norms' means are bounded, the product $\left(I\_d + \frac1n A\_0 \right) \dots \left(I\_d + \frac1n A\_{n-1} \right) $ converges towards $\exp{A}$. We give a dynamical version of this result as well as an illustration with an example of "random walk" on horocycles of the hyperbolic disc.

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