Continuity Properties of Lyapunov Exponents - Lecture 7

Explore the continuity properties of Lyapunov exponents in this lecture from the Simons Semester on Dynamics series. Delve into the control of Lyapunov exponents using entropy for smooth surface diffeomorphisms, based on recent joint works with Sylvain Crovisier and Omri Sarig. Learn about computing the entropy of measures through the expansion of smooth unstable curves using Yomdin's reparametrization technique. Understand the relationship between potential drops in Lyapunov exponents and "near" homoclinic tangencies, which impede the expansion of unstable curves over extended time periods. Gain insights into fundamental concepts in smooth ergodic theory, followed by an examination of key estimates and a discussion of the main result. Time permitting, discover the consequences of this result, including a spectral gap property for entropy-maximizing measures, and explore the construction of Sinai-Ruelle-Bowen measures using these methods.