École d'été finistérienne en systèmes dynamiques

A more precise abstract is given in the PDF file (see below).
I will start to explain concentration inequalities in the context of independent random variables. We will see Azuma-Hoeffding inequality and McDiarmid inequality, and several applications, e.g., to the Kolmogorov-Smirnov test. I will also quickly mention what happens for Markov chains.
Then I will turn to dynamical systems. I will show how to prove a Gaussian concentration bound for a Gibbs measure on a shift of finite type with a Lipschitz potential. In fact, one can prove such an inequality for dynamical systems modeled by a Young tower with a return-time function with exponential tails.
For systems modeled by a Young tower with a return-time function with polynomial tails, we get polynomial concentration bounds. The simplest example is a map of the interval with an indifferent fixed point. I will give several applica- tions of these concentration bounds, for instance to the speed of convergence of the empirical measure toward the SRB measure, or to the almost-sure central limit theorem.

The probabilistic potential theory is the probabilistic counterpart of the classical potential theory, or in other words of the study of harmonic functions. This theory offers the tools to compute transition probabilities for random walks, and for more general Markov chains. For instance, given any finite number of states, we can find the one the Markov chain is most likely to hit first.
We shall present an application of this theory to an elementary example, a simple random walk, and then show how these tools can be adapted to dynamical systems. Along the way, we shall talk about the spectral analysis of transfer operators (and their perturbations), induced systems and Kac's formula, and hitting time of small targets.

Sinai billiards (or the periodic Lorentz gas) are natural dynamical systems which have been challenging mathematicians for half a century. A new tool to study them, Ruelle transfer operators acting on scales of anisotropic Banach spaces, was introduced in 2011 by Mark Demers and Hong-Kun Zhang, who gave a new proof of Lai-Sang Young's celebrated 1998 result of exponential mixing for the SRB measure of the billiard map. I will present results obtained with Mark Demers using this approach, for a natural family of Gibbs states interpolating between the SRB measure and the measure of maximal entropy: For any finite horizon Sinai billiard map T on the two-torus, we find t*>1 such that for each t in (0,t*) there exists a unique equilibrium state for - tlog J^uT, this equilibrium state is exponentially mixing for Holder observables, and the corresponding pressure function P(t) is analytic on (0,t*).

We will investigate invariant measures of stochastic dynamical systems defined by iteration of homeomorphisms of the line.
These types of processes appear naturally in different areas of mathematics. On the one hand they are natural models in probability (more or less applied), on the other hand their behavior in measure is strictly related to the ergodic and geometric properties of the (semi) -groups generated by the transformations.
I will present the results of a recent work in collaboration with D. Buraczewski and T. Szarek, in which we establish conditions (relatively optimal) which guarantee that the system admits a unique invariant measure (possibly of infinite mass).

In the last years, an extremely powerful method has been developed to study the statistical properties of a dynamical system: the functional approach. It consists of the study of the spectral properties of transfer operators on suitable Banach spaces. In this talk I will discuss how to further such a point of view to a class of two dimensional partially expanding maps, not necessarily skew products, in order to provide explicit conditions for the existence of finitely many physical measures and prove exponential decay of correlations for mixing measures. To illustrate the scopes of the theory, I will discuss how to apply the results to a family of fast-slow partially hyperbolic maps. This is a joint work with Carlangelo Liverani.

The generalized semi-standard map is a discrete-time system inspired by Chirikov's standard map. We consider the problem of linearizing the system, i.e conjugating the system analytically to a rotation. We will see that the radius of convergence of the linearization is determined by the Brjuno sum of a multiple of the frequency parameter. This is a joint work with Stefano Marmi.

Given two smooth dynamical systems acting on a compact manifold, one can wonder the extent to which their respective dynamics are similar. The notion of "similarity" strongly depends on the context; in this talk we will focus on the somewhat most constraining version of this question, which is to try and determine when two smooth dynamical systems are conjugate via a differentiable map. This question remains to this day widely open, only a handful of particular cases are well-understood.
We will discuss the standard case of circle diffeomorphisms, whose study stretches over more than a century from Poincaré in 1880 all the way to Yoccoz in the 90s and comprises the work of Denjoy, Arnold, Herman and many others.
We will also review other examples for which differentiable conjugacy classes are partially understood such as the quadratic family z --> z^2 + c and interval exchange maps; and discuss difficulties arising when moving up to dimension 2.
This body of questions in similar in many ways to rigidity questions in Riemannian geometry, such as Mostow or Margulis rigidity, and we will endeavour to highlight links between these two worlds.

We consider a random walk on dxd matrices and the action on the space of flags. Under some conditions, we show that the stationary measure is exact dimensional and give some relations between the dimension, the Furstenberg entropy and the Lyapunov exponents. This is a joint work with Pablo Lessa (Montevideo).

We will discuss two rigidity problems for hyperbolic flows in dimension three. Among smooth Anosov flows on 3-manifolds preserving a smooth volume, algebraic models are believed to distinguish themselves in many ways. The question of entropy rigidity asks whether algebraic models can be characterized (up to smooth conjugacy) by the property that the volume measure has maximal entropy. I will present some results in this direction obtained in a joint project with J. De Simoi, K. Vinhage and Y. Yang. I will also speak about another work with P. Bálint, J. De Simoi and V. Kaloshin in which we were investigating the question of spectral rigidity for a class of dispersing billiards whose dynamics is Axiom A, namely, whether some geometric information can be recovered from the periodic data. In both cases, we shall see that a key step is to obtain precise estimates on the Lyapunov exponents of periodic orbits with a prescribed combinatorics.

A well known approach in order to establish exponential decay of correlations is to prove quasi-compactness of the associated normalized transfer operator. Furthermore, in the presence of singular or parabolic regions, an inducing scheme allows to identify slower rates of decay through an operator renewal theory.
On the other hand, classical models from statistical or quantum physics (like the Dyson-Ising model) do not share this feature of having an isolated singular region which, among other applications, was the motivation to develop tools which allow to prove polynomial decay of correlations without inducing schemes. Furthermore, in order to have direct access to stability, the involved arguments only make use of a projectivization instead of a far more restrictive normalization through eigenfunctions.
This is joint work with Artur Lopes (Porto Alegre, Brazil) and Leandro Cioletti (Brasilia, Brazil).