Continuity Properties of Lyapunov Exponents for Smooth Surface Diffeomorphisms - Lectures 1 and 2

Explore two lectures on continuity properties of Lyapunov exponents for smooth surface diffeomorphisms. Delve into the control of Lyapunov exponents using entropy, following recent joint works by Jérôme Buzzi, Sylvain Crovisier, and Omri Sarig. Learn how to compute the entropy of measures in terms of 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 intervals. Gain insights into fundamental concepts of smooth ergodic theory, followed by main estimates and a thorough discussion of the primary result. If time allows, explore the consequences of this result, including a spectral gap property for entropy-maximizing measures, and the construction of Sinai-Ruelle-Bowen measures. This 1 hour 33 minute lecture series, part of the Simons Semester on Dynamics, is designed for students and researchers in dynamics, even those less familiar with smooth ergodic theory.

Introduction
Plan
Setting
Entropy
Limit
Differential
Natural Blocks
The Plan
The Exercise
High Probability