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Explore advanced concepts in Differentiable Ergodic Theory in 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 through examples and properties of ergodic measures, Bernoulli shifts, and linear torus endomorphisms. Learn about ergodic decomposition theorem, measurable partitions, Rokhlin's disintegration theorem, ergodic uniqueness, and minimality. Cover topics including topological group translations, Haar measure, correlation decay, mixing systems, Markov shifts, ergodic and spectral equivalence, entropy, and the Kolmogorov-Sinai theorem. Additional advanced topics encompass pressure, variational principles, equilibrium states, Ruelle's theorem, exactness and mixing, Hausdorff dimension, and the ergodic theory of non-uniformly hyperbolic systems.
