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Explore advanced concepts in Differentiable Ergodic Theory through this doctoral-level lecture delivered in Portuguese by Professor Marcelo Viana at IMPA. Delve into invariant measures and recurrence, examining Poincaré and Birkhoff Recurrence Theorems, torus rotations, and conservative transformations and flows. Master the weak* topology, von Neumann and Birkhoff ergodic theorems, and subadditive ergodic theorem. Study ergodicity, Bernoulli shifts, torus linear endomorphisms, ergodic decomposition theorem, and Rokhlin's disintegration theorem. Learn about unique ergodicity, minimality, topological group translations, Haar measure, correlation decay, and mixing systems. Investigate Markov shifts, ergodic equivalence, spectral equivalence, entropy concepts including Kolmogorov-Sinai theorem, and topological entropy. Additional topics cover pressure, variational principles, equilibrium states, Ruelle's theorem, exactness and mixing, Hausdorff dimension, and non-uniformly hyperbolic systems theory.
