MATH 6180 - Smooth Ergodic Theory

John Guckenheimer, spring 2015.

From the catalog: Topics include invariant measures; entropy; Hausdorff dimension and related concepts; hyperbolic invariant sets: stable manifolds, Markov partitions and symbolic dynamics; equilibrium measures of hyperbolic attractors; ergodic theorems; Pesin theory: stable manifolds of nonhyperbolic systems; Liapunov exponents; and relations between entropy, exponents, and dimensions.

The course will largely adhere to this description, using the lecture notes by Bowen, Equillibrium States and the Ergodic Theory of Anosov Diffeomorphisms, as reference for the first half of the list of topics. The course will also discuss relevant computational methods for which the book by Kantz and Schreiber, Nonlinear Time Series Analysis, is a reference. Depending upon interest of the class, an alternative topic is to study the work of Artur Avila that was the basis of his 2014 Fields medal.