Learn about equidistribution in this lecture from the Simons Semester on Dynamics series, where Jon Chaika from the University of Utah explores measure preservation and ergodicity. Delve into the Birkhoff ergodic theorem and its implications for continuous compactly supported functions, followed by a proof of ergodicity in irrational circle rotations. Examine Ratner/Masur's argument for ergodicity of flow on the 2-torus, understand Codene's approach to directional flows, and explore hyperbolic total automorphisms through the Hopf argument. Conclude with Dani's perspective on ergodicity of directional flow on the 2-torus, building a comprehensive understanding of these fundamental concepts in dynamical systems.