Differentiable Erg. Theory

Invariant measures and recurrence. Poincaré and Birkhoff recurrence theorems. Rotations on tori. Conservative maps and flows. Existence of invariant measures. Weak* topology. Von Neumann and Birkhoff ergodic theorems. Subadditive ergodic theorem. Ergodicity: examples and properties of ergodic measures. Bernoulli shifts. Linear endomorphisms of the torus. Ergodic decomposition theorem. Mention of measurable partitions and the Rokhlin disintegration theorem. Unique ergodicity and minimality. Translations on topological groups. Haar measure. Decay of correlations. Mixing systems. Markov shifts. Ergodic equivalence and spectral equivalence. Mention of the Ornstein theorem. Entropy. Kolmogorov–Sinai theorem. Mention of the Shannon–McMillan–Breiman theorem. Topological entropy. Subshifts of finite type. Variational principle. Expanding maps on manifolds.

Additional topics: Pressure. Variational principle for the pressure. Expanding transformations on metric spaces. Equilibrium states. Ruelle’s theorem. Exactness and mixing. Hausdorff dimension. Conformal repellers. Hyperbolic attractors and Sinai–Ruelle–Bowen measures. Oseledets’ theorem. Ruelle’s inequality. Pesin’s entropy formula. Ergodic theory of non-uniformly hyperbolic systems.

References:
BOWEN, R. – Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Berlin, Springer-Verlag, 1975.
MAÑÉ, R. – Ergodic Theory and Differentiable Dynamics. Berlin, Springer-Verlag, 1987.
VIANA, M. and OLIVEIRA, K. – Foundations of Ergodic Theory, Cambridge University Press, 2019.