Topics in Ergodic Theory.

Content: Measurable transformations, invariant measures, ergodic theorems --
Lebesgue spaces and measureable partitions, ergodicity and decomposition into ergodic components, spectrum of interval exchange transformations --
Isomorphism of dynamical systems, generators of dynamical systems --
Dynamical systems with pure point spectra --
General properties of eigenfunctions and eigenvalues of ergodic automorphisms, isomorphism of dynamical systems with pure point spectrum --
Entropy theory of dynamical systems --
Breiman theorem, Pinsker partition, K-systems, exact endomorphisms, Gibbs measures --
Entropy of dynamical systems with multidimensional time, systems of cellular automata as dynamical systems --
Continued fractions and Farey fractions --
Homeomorphisms and diffeomorphisms of the circle --
Sharkovski\'s ordering and Feigenbaum\'s universality --
Expanding mappings of the circle --
Standard map, twist maps, periodic orbits, Aubrey-Mather theory --
Periodic hyperbolic points, their stable and unstable manifolds, homoclinic and heteroclinic orbits. Homoclinic and heteroclinic points and stochastic layers --
Geodesic flows and their generalizations, discontinuous dynamical systems, stable and unstable manifolds --
Existence of local manifolds, Gibbs measures --
Markov partitions, [Eta]-theorem for dynamical systems, elements of thermodynamic formalism.