Complexity of Shift Spaces on Semigroups

TL;DR

This paper studies the complexity of shift spaces on semigroups, focusing on topological entropy, and provides methods to compute it, especially for two-symbol cases, extending previous work on free semigroups.

Contribution

It establishes the existence of topological entropy for G-shift of finite type and characterizes it for two-symbol semigroups, extending prior results.

Findings

Topological entropy exists for G-shift of finite type.

Entropy calculation reduces to solving nonlinear recurrence equations.

Complete characterization of entropy for two-symbol G-shifts.

Abstract

Let $G=\left\langle S|R_{A}\right\rangle $ be a semigroup with generating set $ S$ and equivalences $R_{A}$ among $S$ determined by a matrix $A$. This paper investigates the complexity of $G$-shift spaces by yielding the topological entropies. After revealing the existence of topological entropy of $G$-shift of finite type ($G$-SFT), the calculation of topological entropy of $G$-SFT is equivalent to solving a system of nonlinear recurrence equations. The complete characterization of topological entropies of $G$-SFTs on two symbols is addressed, which extends [Ban and Chang, arXiv:1803.03082] in which $G$ is a free semigroup.