Minicourse #1
Title: Measure rigidity in hyperbolic dynamics
Speaker: Davi Obata (Brigham Young University, USA)
Abstract: Understanding the behavior of the orbits of a general dynamical system is a very hard task. Another approach is to try to understand the average behavior of many orbits. This gives the idea of physical measures, measures that capture the statistical behavior of a set of points having positive volume. An important problem in smooth dynamics is to understand mechanisms that imply the existence of physical measures.
In this mini-course, we will explore the relation between invariant foliations (stable/unstable foliations), certain types of invariant measures (u-Gibbs/SRB), and how this is connected with the problem of finding mechanisms for the existence of physical measures. We will explore techniques to show rigidity of certain types of measures for Anosov systems.
Mincourse #2
Title: The marked length spectrum of Anosov surfaces
Speaker: Thibault Lefeuvre (CNRS / Sorbonne Université, France)
Abstract: Anosov surfaces are closed Riemannian surfaces whose geodesic flow is Anosov, such as those of negative curvature. The marked length spectrum is the data of the lengths of closed geodesics, marked by the free homotopy of the surface. The purpose of this minicourse is to prove the following rigidity theorem: the marked length spectrum of Anosov surfaces determines the metric (up to isometries). The proof involves various techniques, including hyperbolic dynamics, Riemannian geometry, and microlocal analysis. I will take some time to introduce all the necessary tools — no prerequisites required!