Computability of Pressure for Subshifts on Countable Amenable Groups

TL;DR

This paper proves that topological pressure for subshifts on countable amenable groups is computable under certain conditions, extending previous results and providing explicit bounds on computation time, especially for $ ext{Z}^d$-subshifts.

Contribution

It establishes the computability of topological pressure for a broad class of subshifts on amenable groups, generalizing prior results limited to $ ext{Z}^d$-subshifts.

Findings

Topological pressure is computable from above given a forbidden list.

For strongly irreducible shifts of finite type, pressure is fully computable.

Explicit bounds on computation time are provided, with singly exponential time for $ ext{Z}^2$-subshifts.

Abstract

There are a variety of results in the literature proving forms of computability for topological entropy and pressure on subshifts. In this work, we prove two quite general results, showing that topological pressure is always computable from above given an enumeration for a forbidden list inducing the subshift, and that for strongly irreducible shifts of finite type, topological pressure is computable. Our results apply to subshifts on all finitely generated amenable groups with decidable word problem and generalize several previous results which applied only to $\mathbb{Z}^d$-subshifts. As corollaries, we obtain some results related to ground state energy and entropy, proving that the map sending $\phi$ to $\sup_{\mu \in M_\sigma(X)} \int \phi d\mu$ is computable/computable from above when $P_X(\phi)$ is, and that the map sending $\phi$ to its ground state/residual entropy is computable from above when $P_X(\phi)$ is computable. We conclude by giving explicit bounds on computation time of $P_X(\phi)$ in the $\mathbb{Z}^d$ setting for SI SFTs and locally constant and rational valued $\phi$, and show that in the special case $X = A^{\mathbb{Z}^2}$, this algorithm runs in singly exponential time.