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 and recurrence, 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, and subadditive ergodic theorem. Examine ergodicity through examples and properties of ergodic measures, Bernoulli shifts, and linear torus endomorphisms. Study ergodic decomposition theorem, measurable partitions, Rokhlin's disintegration theorem, ergodic uniqueness, and minimality. Learn about topological group translations, Haar measure, correlation decay, mixing systems, Markov shifts, ergodic and spectral equivalence, entropy concepts including Kolmogorov-Sinai theorem, and expanding transformations on manifolds. Additional advanced topics cover pressure, variational principles, equilibrium states, Ruelle's theorem, exactness and mixing, Hausdorff dimension, conformal repellers, hyperbolic attractors, and non-uniformly hyperbolic systems theory. Read more