(11/26/2024) - PhD: Differentiable Ergodic Theory - Marcelo Viana - Lecture 26
Professor: Marcelo Viana
Playlist of videos: https://bit.ly/4dbuEzl
Invariant measures and recurrence. Poincaré and Birkhoff recurrence theorems. Rotations on tori. Conservative transformations 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 Rokhlin's decay theorem. Ergodic uniqueness and minimality. Translations in topological groups. Haar measure. Decay of correlations. Mixing systems. Markov shifts. Ergodic equivalence and spectral equivalence. Mention of Ornstein's theorem. Entropy. Kolmogorov–Sinai theorem. Mention of Shannon–McMillan–Breiman theorem. Topological entropy. Finite-type displacements. Variational principle. Expanding transformations on manifolds.
Additional topics: Pressure. Variational principle for 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. – Fundamentals of Ergodic Theory, 2nd edition, Rio de Janeiro, Brazilian Mathematical Society, 2019.
IMPA Social Networks: https://linktr.ee/impabr
IMPA - Institute of Pure and Applied Mathematics ©
https://www.impa.br | https://impa.br/videos
The rights to all material on this channel belong to the Institute of Pure and Applied Mathematics, and the total or partial use of the content is prohibited without prior written authorization from the aforementioned owner, except in the cases provided for in current legislation.
The rights over all the material in this channel belong to the Instituto de Matemática Pura e Aplicada, and it is prohibited to use all or part of it without prior written authorization from the above mentioned holder, except in the cases prescribed in the current legislation.