# Analysis Seminar-Complexity, periodicity, and applications to zero entropy symbolic dynamics

**Speaker:** Van Cyr, Bucknell University

**Date:** Thursday, March 3 at 2:00 PM

**Where:** Room 771

**Title:** Complexity, periodicity, and applications to zero entropy symbolic dynamics

Abstract: The automorphism group of a symbolic dynamical system (X; ) is the group of homeomorphisms of X that commute with . For many natural systems, this group is extremely large and complicated (e.g. a theorem of Boyle, Lind, and Rudloph shows that if X is a topologically mixing SFT, then Aut(X) contains isomorphic copies of all nite groups, the free group on two generators, and the direct sum of countably many copies of **Z**). This can be interpreted as a manifestation of the\high complexity" of these shifts. In this talk I will discuss recent joint work with B. Kra which places restrictions on the automorphism group of any subshift of \low complexity." This class is quite general and contains any minimal subshift whose factor complexity function has stretched exponential growth (with stretching exponent < 1=2). I will also discuss how the same proof techniques can be used to study the probability measures on X that are invariant under a subgroup of Aut(X).