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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, and unique ergodicity and minimality. Cover advanced topics including topological entropy, finite-type shifts, variational principle, and expanding transformations on manifolds, with additional focus on pressure, equilibrium states, Ruelle's theorem, and non-uniformly hyperbolic systems.
