Mathematics

Abstract ergodic theory is the study of measurable measure preserving transformations from a probabilistic/statistical point of view. In general however a differentiable dynamical system is defined on a manifold with a reference measure (e.g. the Riemannian volume) which is not a priori invariant for the dynamics. In such a situation, the first step for the application of the powerful methods of ergodic theory, is to study the existence (or not) of invariant measures. Of particular interest are so-called "Physical Measures" such as ergodic measures which are absolutely continuous with respect to the Riemannian volume. 

 

In this course I will give a gentle introduction to basic concepts of ergodic theory such as invariant measures and ergodicity, give several examples, prove some basic results such as Birkhoff's ergodic theorem, and finally prove the existence of absolutely continuous invariant measures for a class of one-dimensional maps.