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Explore the continuity properties of Lyapunov exponents in two lectures from the Simons Semester on Dynamics. 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 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 of smooth ergodic theory, followed by main estimates and a thorough discussion of the primary result. Time permitting, explore consequences such as the spectral gap property for entropy-maximizing measures and the construction of Sinai-Ruelle-Bowen measures.
