Professor Omri Sarig: Measures of Maximal Entropy for C Infinity Surface Diffeomorphisms

Date: 2024-11-12

Time: 16:10 - 17:00

Speaker
Professor Omri Sarig, Weizman Institute of Science

Abstract

Let f be a C infinity surface diffeomorphism with positive topological entropy. Newhouse showed that f admits measures of maximal entropy (MME), and Buzzi, Crovisier and I showed that the number of ergodic MMEs is finite.

The main result of the talk is that for each ergodic MME there is a L.-S. Young tower extension of a set of full measure, with the following properties:

(1) the map from the tower to the manifold is almost-surely finite-to-one;

(2) the tail of the first return time map, with respect to the MME, decays exponentially.

By results of L.-S. Young and others, this implies a variety of properties for the MME, including exponential mixing, the almost sure invariance principle, and quantitative bounds for the Lyapunov exponents of measures with nearly maximal entropy. 

This is joint work with Jerome Buzzi and  Sylvain Crovisier.