Colloquium: Piotr Oprocha

Speaker: Piotr Oprocha, AGH University of Science and Technology, Krakow, Poland 

Title: On the shadowing property and odometers

Abstract: When we investigate the space of invariant measures from ergodic theory point of view, we are usually not that much interested in the topological structure of the underlying space. By the famous Jewett-Krieger theorem, we can view invariant measures as supported on minimal systems. Numerous further generalizations allow to add even more topological (dynamical) properties to the underlying system. On the other hand, there are examples of systems with quite rich dynamical structure (e.g. topologically mixing) but not that much interesting invariant measures (e.g. only trivial, only atomic, etc.). In other words, connections between topology and ergodic theory (on compact metric spaces) is not that tight.

In this talk we will provide some characterizations of invariant measures in the case when a dynamical system $(X,T)$ has shadowing property. We will show that often invariant measures can be approximated by a special class of minimal dynamical systems. We will also comment on possibilities of approximation of entropy.

This is joint work with Jian Li.

Faculty host: Krystyna Kuperberg