Christian Wolf : Computability of topological entropy and pressure for symbolic systems beyond finite type

Posted by Shmuel Friedland , part of the Departmental Colloquium.

At

Oct. 25, 2024, 3 p.m.

In

636 SEO

Abstract

The computability of dynamically defined objects has been a subject of intensive study over the past two decades. This includes important results concerning the computability of invariant sets (e.g. Julia sets), entropies, natural invariant measures, Lyapunov exponents, etc. In this talk we consider symbolic dynamical systems given by the shift map on a finite alphabet-shift space $X$. We identify the computability of the topological entropy $h_{\rm top}(X)$ and topological pressure $P_{\rm top}(X,\phi)$ as a function of the shift space $X$ and potential $\phi$. This question has previously been studied by Burr et al., Hertling and Spandl, and Spandl in some special cases. In this talk we address the computability question for general shift spaces. One of our results states that the entropy function $X\mapsto h_{\rm top}(X)$ is computable at a shift space $X_0$ iff $X_0$ has zero topological entropy.