Teoria Ergódica Diferenciável - Aula 26

Explore advanced concepts in Differentiable Ergodic Theory through this doctoral-level lecture delivered in Portuguese by Professor Marcelo Viana at IMPA (Instituto de Matemática Pura e Aplicada). Delve into key topics including invariant measures, Poincaré and Birkhoff recurrence theorems, torus rotations, conservative transformations and flows, and the existence of invariant measures. Master weak* topology, von Neumann and Birkhoff ergodic theorems, subadditive ergodic theorem, and ergodicity with examples and properties of ergodic measures. Study Bernoulli shifts, torus linear endomorphisms, ergodic decomposition theorem, measurable partitions, Rokhlin's disintegration theorem, ergodic uniqueness, minimality, and topological group translations. Examine Haar measure, correlation decay, mixing systems, Markov shifts, ergodic and spectral equivalence, Ornstein's theorem, entropy, Kolmogorov-Sinai theorem, Shannon-McMillan-Breiman theorem, topological entropy, finite type shifts, variational principle, and expanding transformations on manifolds. Additional advanced topics cover pressure, 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.