Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## Symbolic systems, automorphism groups, and invariant measures

Talk by Prof. Dr. Bryna Kra

**Date:** 12.04.21 **Time:** 16.00 - 17.15 **Room:** ETH HG G 19.2

The automorphism group of a shift system can be quite complicated: for example, for a topologically mixing shift of finite type, the automorphism group contains isomorphic copies of all finite groups and the free group on two generators and such behavior is common for shifts of high complexity. In the opposite setting of low complexity, there are numerous restrictions on the automorphism group, and for many classes of shift systems, it is known to be virtually abelian. In both of these extreme cases, there exists a Borel probability measure supported on the shift that is invariant under the full automorphism group. However, it unknown if every shift system supports such a measure. I will discuss progress on this question, including its solution for a new class of shift systems, which we have named the language stable shifts. This is joint work with Van Cyr.