Teoria Ergódica Diferenciável - Aula 03 - 20/08/2024

Explore a comprehensive doctoral-level lecture on Differentiable Ergodic Theory delivered by Professor Marcelo Viana at the Instituto de Matemática Pura e Aplicada. Delve into advanced topics such as invariant measures, recurrence, Poincaré and Birkhoff recurrence theorems, torus rotations, conservative transformations and flows, and the existence of invariant measures. Examine weak* topology, von Neumann and Birkhoff ergodic theorems, subadditive ergodic theorem, ergodicity, Bernoulli shifts, and linear endomorphisms of the torus. Study the ergodic decomposition theorem, measurable partitions, Rokhlin's disintegration theorem, unique ergodicity, minimality, and translations in topological groups. Investigate Haar measure, decay of correlations, mixing systems, Markov shifts, ergodic and spectral equivalence, and Ornstein's theorem. Explore entropy, the Kolmogorov-Sinai theorem, Shannon-McMillan-Breiman theorem, topological entropy, finite type shifts, and the variational principle. Conclude with expanding transformations on manifolds and additional advanced topics in ergodic theory.