Computability of topological pressure on compact shift spaces beyond finite type

TL;DR

This paper explores the limits of computability for topological pressure in various shift spaces, establishing new results for specific classes and revealing cases where it is non-computable, thus advancing understanding in symbolic dynamics and computable analysis.

Contribution

It develops a framework for analyzing the computability of topological pressure on general shift spaces and applies it to several classes, including coded shifts and Beta-shifts, showing both computability and non-computability results.

Findings

Computability of topological pressure is established for S-gap, generalized gap, and certain Beta-shifts.

Constructed shift spaces where pressure is either computable or non-computable depending on the potential.

The entropy map is computable if and only if the shift space has zero topological entropy.

Abstract

We investigate the computability (in the sense of computable analysis) of the topological pressure $P_{\rm top}(\phi)$ on compact shift spaces $X$ for continuous potentials $\phi:X\to {\mathbb R}$. This question has recently been studied for subshifts of finite type (SFTs) and their factors (Sofic shifts). We develop a framework to address the computability of the topological pressure on general shift spaces and apply this framework to coded shifts. In particular, we prove the computability of the topological pressure for all continuous potentials on S-gap shifts, generalized gap shifts, and particular Beta-shifts. We also construct shift spaces which, depending on the potential, exhibit computability and non-computability of the topological pressure. We further prove that the generalized pressure function $(X,\phi)\mapsto P_{\rm top}(X,\phi\vert_{X})$ is not computable for a large set of shift spaces $X$ and potentials $\phi$. In particular, the entropy map $X\mapsto h_{\rm top}(X)$ is computable at a shift space $X$ if and only if $X$ has zero topological entropy. Along the way of developing these computability results, we derive several ergodic-theoretical properties of coded shifts which are of independent interest beyond the realm of computability.