Graph subshifts

TL;DR

This paper introduces graph subshifts of finite type, extending classical symbolic dynamics and group theory, and investigates their properties related to local pattern constraints and periodicity.

Contribution

It defines a new class of graph subshifts of finite type and analyzes their capacity to enforce specific support graphs, linking to classical dynamics.

Findings

Subshifts with only infinite graphs are either aperiodic or lack residual finiteness.

The paper presents non-trivial examples of such subshifts.

Two undecidability theorems are established for these models.

Abstract

We propose a definition of graph subshifts of finite type that can be seen as extending both the notions of subshifts of finite type from classical symbolic dynamics and finitely presented groups from combinatorial group theory. These are sets of graphs that are defined by forbidding finitely many local patterns. In this paper, we focus on the question whether such local conditions can enforce a specific support graph, and thus relate the model to classical symbolic dynamics. We prove that the subshifts that contain only infinite graphs are either aperiodic, or feature no residual finiteness of their period group, yielding non-trivial examples as well as two natural undecidability theorems.