Dynamics and entropy of $\mathcal{S}$-graph shifts

TL;DR

This paper introduces $\\mathcal{S}$-graph shifts, a new class of shift spaces represented by finite directed graphs with numerical labels, unifying various existing shift types and providing explicit formulas for entropy and zeta functions.

Contribution

It defines $\\mathcal{S}$-graph shifts, generalizing $S$-gap shifts and other shift spaces, and derives formulas for their entropy and zeta functions, solving a previously posed problem.

Findings

Derived a formula for the entropy of $\\mathcal{S}$-graph shifts.

Established an explicit formula for their zeta functions.

Showed that all entropy values are realized by uncountably many such shifts.

Abstract

$S$-gap shifts are a well-studied class of shift spaces, which has led to several proposed generalizations. This paper introduces a new class of shift spaces called $\mathcal{S}$-graph shifts whose essential structure is encoded in a novel way, as a finite directed graph with a set of natural numbers assigned to each vertex. $\mathcal{S}$-graph shifts contain $S$-gap shifts and their generalizations, as well as all vertex shifts and SFTs, as special cases, thereby providing a method to study these shift spaces in a uniform way. The main result in this paper is a formula for the entropy of any $\mathcal{S}$-graph shift, which, by specialization, resolves a problem proposed by Matson and Sattler. A second result establishes an explicit formula for the zeta functions of $\mathcal{S}$-graph shifts. Additionally, we show that every entropy value is obtained by uncountably many $\mathcal{S}$-graph shifts.