Searcharxiv⌕ Search

arXiv subjects

Charles Fougeron

Publications and source records attributed to Charles Fougeron.

12 recordsLinked to original sources
arXiv2026

Rauzy-Veech Induction for Infinite-Type IETs

We consider infinite-type IETs arising from elementary examples of finite-area translation surfaces of infinite genus such as the Baker's surface. We call such IETs tail-reversing and we show that for any tail-reversing permutation the subset of the simplex of lengths $\Delta$ for which the corresponding infinite-type IET is uniquely ergodic contains a dense $G_{\delta}$ set with respect to the $\ell^1$-topology. To this end, we generalize Rauzy-Veech induction to a large class of infinite-type IETs, where we prove a minimality criterion as a generalization of Keane's criterion in the finite setting. We then restrict ourselves to tail-reversing IETs and obtain our genericity result through a combinatorial analysis of their infinite-type Rauzy diagrams. Moreover, we derive an explicit condition for a tail-reversing IET to be uniquely ergodic by studying the diameter of its induction matrices.

math.DS↗
arXiv2025

Dynamical decomposition of generalized interval exchange transformations

We develop a renormalization scheme which extends the classical Rauzy-Veech induction used to study interval exchange tranformations (IETs) and allows to study generalized interval exchange transformations (GIETs) $T: [0,1) \to [0,1)$ with possibly more than one quasiminimal component (i.e. not infinite-complete, or, equivalently, not semi-conjugated to a minimal IET). The renormalization is defined for more general maps that we call interval exchange transformations with gaps (g-GIETs), namely partially defined GIETs which appear naturally as the first return map of $C^r$-flows on two-dimensional manifolds to any transversal segment. We exploit this renormalization scheme to find a decomposition of $[0,1)$ into finite unions of intervals which either contain no recurrent orbits, or contain only recurrent orbits which are closed, or contain a unique quasiminimal. This provides an alternative approach to the decomposition results for foliations and flows on surfaces by Levitt, Gutierrez and Gardiner from the $1980s$.

math.DS↗
arXiv2024

On asymptotic expansions of ergodic integrals for $\Z^d$-extensions of translation flows

We obtain expansions of ergodic integrals for $\Z^d$-covers of compact self-similar translation flows, and as a consequence we obtain a form of weak rational ergodicity with optimal rates. As examples, we consider the so-called self-similar $(s,1)$-staircase flows ($\Z$-extensions of self-similar translations flows of genus-$2$ surfaces), and particular cases of the Ehrenfest wind-tree model.

math.DS↗
arXiv2021

A cyclotomic family of thin hypergeometric monodromy groups in ${Sp}_4(\mathbb{R})$

We exhibit an infinite family of discrete subgroups of ${Sp}_4(\mathbb R)$ which have a number of remarkable properties. Our results are established by showing that each group plays ping-pong on an appropriate set of cones. The groups arise as the monodromy of hypergeometric differential equations with parameters $\left(\tfrac{N-3}{2N},\tfrac{N-1}{2N}, \tfrac{N+1}{2N}, \tfrac{N+3}{2N}\right)$ at infinity and maximal unipotent monodromy at zero, for any integer $N\geq 4$. Additionally, we relate the cones used for ping-pong in $\mathbb R^4$ with crooked surfaces, which we then use to exhibit domains of discontinuity for the monodromy groups in the Lagrangian Grassmannian.

math.AG↗
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.

math.DS↗
arXiv2020

Dynamical properties of simplicial systems and continued fraction algorithms

We propose a new point of view on multidimensional continued fraction algorithms inspired by Rauzy induction. The generic behaviour of such an algorithm is described here as a random walk on a graph that we call simplicial system. These systems provide a family of examples for random walks with memory recorded by a finite dimensional vector. We introduce a general criterion on these graphs that induces ergodicity together with a bundle of many other dynamical properties. In particular, after computing the representation of Brun, Selmer and Arnoux-Rauzy-Poincare algorithm in this formalism, it provides a unified proof of ergodicity for these classical examples as well as new results such as uniqueness of the measure of maximal entropy on a canonical suspension. These objects also bring a new perspective to some fractal sets such as Rauzy gaskets. We show general explicit upper bound on Hausdorff dimensions of fractals described in this formalism as well as a construction of their measure of maximal entropy. This implies in particular that the Rauzy gasket in all dimensions has Hausdorff dimension strictly smaller than its ambient space, as well as sharper bounds on the dimension and an asymptotic result.

math.DS↗
arXiv2019

Cascades in the dynamics of affine interval exchange transformations

We describe in this article the dynamics of a $1$-parameter family of affine interval exchange transformations. It amounts to studying the directional foliations of a particular affine surface, the Disco surface. We show that this family displays various dynamical behaviours: it is generically dynamically trivial, but for a Cantor set of parameters the leaves of the foliations accumulate to a (transversely) Cantor set. s study is achieved through the analysis the dynamics of the Veech group of this surface combined a modified version of Rauzy induction in the context of affine interval exchange transformations.

math.DS↗
arXiv2018

Diffusion rate of windtree models and Lyapunov exponents

Consider a windtree model with several parallel arbitrary right-angled obstacles placed periodically on the plane. We show that its diffusion rate is the largest Lyapunov exponent of some stratum of quadratic differentials and exhibit a new general strategy to compute the generic diffusion rate of such models. This result enables us to compute numerically the diffusion rates of a large family of models, and to observe its asymptotic behaviour according to the shape of the obstacles.

math.DS↗
arXiv2017

Lyapunov exponents of the Hodge bundle over strata of quadratic differentials with large number of poles

We show an upper bound for the sum of positive Lyapunov exponents of any Teichmüller curve in strata of quadratic differentials with at least one zero of large multiplicity. As a corollary, it holds for any $SL(2,\mathbb R)$-invariant subspaces defined over $\mathbb Q$ in these strata. This proves Grivaux-Hubert's conjecture about the asymptotics of Lyapunov exponents for strata with a large number of poles in the situation when at least one zero has large multiplicity.

math.DS↗
arXiv2017

Parabolic degrees and Lyapunov exponents for hypergeometric local systems

Consider the flat bundle on $\mathrm{CP}^1 - \{0,1,\infty \}$ corresponding to solutions of the hypergeometric differential equation $ \prod_{i=1}^h (\mathrm D - α_i) - z \prod_{j=1}^h (\mathrm D - β_j) = 0$ where $\mathrm D = z \frac {d}{dz}$. For $α_i$ and $β_j$ distinct real numbers, this bundle is known to underlie a complex polarized variation of Hodge structure. Setting the complete hyperbolic metric on $\mathrm{CP}^1 - \{0,1,\infty \}$, we associate $n$ Lyapunov exponents to this bundle. We compute the parabolic degrees of the holomorphic subbundles induced by the variation of Hodge structure and study the dependence of the Lyapunov exponents in terms of these degrees by means of numerical simulations.

math.AG↗
arXiv2016

Affine surfaces and their Veech groups

We introduce a class of objects which we call 'affine surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in order to motivate several questions and open problems. In particular we generalise the notion of Veech group to affine surfaces, and we prove a structure result about these Veech groups.

math.GT↗