Conference Probability and Dynamics

Let (X_i) be a sequence of iid GL_d(R)-valued random matrices (where the dimension d is being fixed once for all). The (non commutative) random walk S_n=X_n ... X_1 on the group GL_d(R) induces a Markov chain on the projective space P(R^d). The goal of the talk is to give a full (qualitative) classification of the stationary probability measures of this Markov chain without any irreducibility assumption, generalizing results of Bougerol--Picard (92) concerning affine random walks and linking the works of Furstenberg--Kifer (82) and those Guivarch--Raugi (07) and Benoist--Quint (16) in the reductive case. Joint work with Çağri Sert.

We discuss the recurrence properties of the model indicated in the title. The analysis involves former results of Jean-Pierre Conze and Yves Guivarc'h (2000) on random walks in random medium and quasi-invariant measures, that we shall recall precisely.

Random products of matrices have been widely studied in the last years for both their theoretical interest and various application. We now have a rich and complete theory for random walks on linear group, however less is known when the matrices are not supposed to be invertible.
The goal of this talk is to present through the exemple of the semigroup of rank one matrices the interest of exploring random walks on semigroups, presenting reasonable generalisation of the group case, new challenges and some results.

Bowen and Margulis independently proved in the 70s that closed geodesics on compact hyperbolic surfaces equidistribute towards the measure of maximal entropy. From a homogeneous dynamics point of view, this measure is the quotient of the Haar measure on PSL(2,R) modulo some discrete cocompact sugroup. In a joint work with Jialun Li, we investigate the higher rank setting of this problem by taking a higher rank Lie group (like SL(d,R) for d>2) and by studying the dynamical properties of geodesic flows in higher rank : the so-called Weyl chamber flows and their induced diagonal action. We obtain an equidistribution formula of periodic tori (instead of closed orbits of the geodesic flow).

We compute the tau-mixing coefficients of a class of dynamical systems. We give some applications to deviations inequalities in case of polynomial or sub-exponential decay of correlations. Based on a joint work with C. Cuny and F. Merlevède

We prove the Central Limit Theorem (CLT), the first order Edgeworth Expansion and a Mixing Local Central Limit Theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise smooth expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on the real line. The class of observables in the CLT and the MLCLT on the real line include the real part, the imaginary part and the absolute value of the Riemann zeta function. Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of the results of Lifschitz & Weber (2009) and Steuding (2012) who have proven the Strong Law of Large Numbers for sampling the Lindelöf hypothesis.

The talk will consists of a long historical introduction to the topic of deviation of ergodic averages for locally Hamiltonian flows on compact surafces as well as some current results obtained in collaboration with Corinna Ulcigrai and Minsung Kim. New developments include a better understanding of the asymptotic of so-called error term (in non-degenerate regime) and the appearance of new exponents in the deviation spectrum (in degenerate regime).

We describe a general approach to the theory of self consistent transfer operators. These operators have been introduced as tools for the study of the statistical properties of a large number of all to all interacting dynamical systems subjected to a mean field coupling. We consider a large class of self consistent transfer operators and prove general statements about existence of invariant measures, speed of convergence to equilibrium, statistical stability and linear response. While most of the results presented are valid in a weak coupling regime, the existence results for the invariant measures we show also hold outside the weak coupling regime. We also consider the problem of finding the optimal coupling between maps in order to change the statistical properties of the system in a prescribed way.

We study transfer operators associated to piecewise monotone interval transformations and show that the essential spectrum is large whenever the Banach space bounds L^\infty and the transformation fails to be Markov. Constructing a family of Banach spaces we show that the lower bound on the essential spectral radius is optimal. Indeed, these Banach spaces realise an essential spectral radius as close as desired to be the theoretical best possible case.

Strictly convex divisible sets X are the universal covers of real projective compact manifolds M. From the dynamical point of view, these objects yield natural examples of topologically mixing Anosov flows. The boundary of a strictly convex divisible set is not very smooth in general: unless X is conical, i.e., an ellipsoid (and, thus, M is a hyperbolic manifold), its boundary is not C^2. In this talk, based on joint work with P. Foulon and P. Hubert, we shall discuss the infinitesimal bends at a typical point of the boundary of a non-conical strictly convex divisible set: in particular, we shall see that these infinitesimal bends are highly anisotropic (i.e., they are distinct in distinct directions) thanks to the features of certain locally constant cocycles over uniformly hyperbolic systems.

An important question in stochastic analysis is the appropriate interpretation of stochastic integrals. The classical Wong-Zakai theorem gives sufficient conditions under which smooth integrals converge to Stratonovich stochastic integrals. The conditions are automatic in one-dimension, but in higher dimensions it is necessary to take account of corrections stemming from the Levy area.
In previous work with Kelly, we justified the Levy area correction for large classes of deterministic systems, bypassing any stochastic modelling assumptions.
This talk addresses a much less studied question: is the Levy area zero or nonzero for systems of physical interest, eg Hamiltonian time-reversible systems? In recent work with Gottwald, we classify (and clarify) the situations where such structure forces the Levy area to vanish. Outside of these rare situations, we show that the Levy area is typically nonzero.

For a centered random walk S_n on R, given x in R, one can estimate the probability that x+S_n belongs to an interval [a,b], while all the x+S_k, 0< k< n, have remained non-negative: it decays as n^{-3/2}. Following a general principle which one can date back to Sinai, this should have an analogue for Birkhoff sums of Hölder continuous observables over hyperbolic dynamical systems. Indeed, this is the case, as follows from our joint work with Ion Grama and Hui Xiao.

In this talk, we are going to take interest in the following general problem. Let L be a unimodular lattice of R^{d}. Let S be a measurable set of finite volume. What is the number N(S,L) of points that belong to both L and S? When S is sufficiently regular, it can be shown that this number N(S,L) is approximated by vol(S) to within an error denoted by R(S,L). The aim of this talk is to present several results that gives us a better idea of the behaviour of R(S,L) in different situations: for example, when S is the united disk dilated by a factor t with t going to infinity and L being a random unimodular lattice or when S is a parallelogram dilated by a factor t with t going to infinity and L being a random unimodular lattice.

Billiards are well-known models first introduced by Birkhoff as paradigmatic examples of Hamiltonian systems, and pioneered by Yakov Sinai, Leonid Bunimovich, Nikolai Chernov, etc., as a mathematical model for the Lorenz systems and hard-ball gases. Since then billiards have acquired increasing importance as they shed light in understanding thermodynamic limits, connected to deep issues in quantum and wave physics all the way to quantum chaos. However for some Bunimovich billiards, whose boundary consists of only arcs and straight lines, even the hyperbolicity is not known, not to mention ergodicity and other chaotic properties. In this talk, I will discuss these new classes of convex billiards and their properties. The hyperbolicity for some of them is only numerically proved, which leaves many open questions to explore. I will also review my recent collaboration work with Michal Misiurewicz about topological entropy for Bunimovich stadium.