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

Travis Dillon

doi:10.3934/dcds.2022117

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2022, Volume 42, Issue 12: 5637-5663. Doi: 10.3934/dcds.2022117

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

Travis Dillon^1,

1.
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received: September 2021

Revised: February 2022

Early access: August 2022

Published: December 2022

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Abstract

$ S $-gap shifts are a well-studied class of shift spaces which have 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.

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Mathematics Subject Classification: Primary: 37B10.

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