Teoria Ergódica Diferenciável - Aula 02 - 15/08/2024

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, and subadditive ergodic theorem. Investigate ergodicity, Bernoulli shifts, linear endomorphisms of the torus, ergodic decomposition theorem, and unique ergodicity. Study translations in topological groups, Haar measure, decay of correlations, mixing systems, and Markov shifts. Explore ergodic and spectral equivalence, entropy, Kolmogorov-Sinai theorem, topological entropy, and expanding transformations on manifolds. Additional topics include pressure, variational principle, equilibrium states, Ruelle's theorem, exactness and mixing, Hausdorff dimension, conformal repellers, hyperbolic attractors, Sinai-Ruelle-Bowen measures, Oseledets theorem, Ruelle inequality, Pesin entropy formula, and ergodic theory of non-uniformly hyperbolic systems.