Dylan BANSARD-TRESSE

Research interests

Dynamical systems, ergodic theory, concentration, extreme value theory, point processes

Concentration inequalities allow to control the fluctuations of very general observables, in particular those which are implicitly defined. Ergodic sums are of course included as a very special case. There are many different applications of these inequalities, some being of probabilistic nature, other ones of statistical nature. For instance, speed of convergence of the empirical distance to the invariant measure, empirical entropies, estimation of correlation dimension, etc. In the realm of dynamical systems defined by the iterations of a non-uniformly hyperbolic map, there are important results already known but there are still many open questions. A first series of questions is about Markov shifts on a countably infinite alphabet. They form a very interesting class notably displaying phase transitions (non-uniqueness of equilibirum states). The first goal of this thesis is to obtain a complete classification of the different types of concentration behaviors in terms of the regularity of the potential. They range from Gaussian to polynomial. The second goal is to study in depth other nonuniformly hyperbolic dynamical systems for which one can play with two sources of nonuniform hyperbolicity: decay of the tail of return times to the reference set which allows to regain some hyperbolicity, and regularity of the abstract Jacobian. One goal is to obtain optimal conditions under which one still has Gaussian concentration. Finally, on the side of applications, there are new observables to deal with, especially the Kantorovich distance between the empirical measure and the invariant measure of the system (SinaiRuelle-Bowen measure). Indeed, one has to control its expected value which is well known and easy only in dimension one. For higher dimensional dynamical systems with a strange atractor, it is an open question to get a sharp bound.