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

\(S\)-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 nu...

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Bibliographic Details
Published in:arXiv.org 2022-07
Main Author: Dillon, Travis
Format: Article
Language:English
Subjects:
Dynamical systems
Entropy
Graph theory
Online Access:Get full text
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Summary:\(S\)-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.
ISSN:2331-8422