Differentiable Ergodic Theory

Poincaré’s recurrence theorem and Birkhoff’s ergodic theorem. Existence of invariant measures for continuous transformations. Ergodic and mixing transformations. Uniquely ergodic transformations; examples: shifts, automorphisms and trunk translation. Ergodic decomposition of invariant measures. Metric and topological entropy. Expanding transformations and the existence of variational measures. Additional topics: variational principle; equilibrium states; hyperbolic attractors and Sinai-Ruelle-Bowen measures. Oseledec’s theorem. Ruelle’s inequality and Pesin’s entropy formula. Ergodic theory of non-uniform hyperbolic systems.

References:
BOWEN, R. – Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Berlin, Springer-Verlag, 1975.
CRAIZER, M. – Entropia das Funções Internas. Rio de Janeiro, IMPA, 1989.
MAÑÉ, R. – Ergodic Theory and Diferentiable Dynamics. Berlin, Springer-Verlag, 1987.

* Standard program. The teacher has the autonomy to make any changes.