Stretching and Shrinking - 85 Years of the Hopf Argument for Ergodicity

Watch a comprehensive mathematics lecture exploring the historical development and applications of the Hopf argument for ergodicity, from its origins in early 20th century Harvard to modern applications. Delve into how Gustav Hedlund's 1934 proof of geodesic flow ergodicity for closed hyperbolic surfaces led to Eberhard Hopf's groundbreaking 1939 proof introducing the Hopf argument - a "soft" method for demonstrating ergodicity in hyperbolic systems. Examine three major theoretical results that employ the Hopf argument: the Hopf-Anosov theorem on geodesic equidistribution in negatively curved manifolds, Mostow's rigidity theorem for hyperbolic manifolds, and the Mañé-Avila-Crovisier-Wilkinson theorem on ergodicity of generic symplectomorphisms. Learn how this fundamental concept connects early work on Boltzmann's Ergodic Hypothesis with modern developments in geometry, representation theory, and dynamical systems.