Four-Cycle Free Graphs, Height Functions, the Pivot Property and Entropy Minimality

TL;DR

This paper studies vertex shift spaces derived from four-cycle free graphs, proving they have the pivot property and are entropy minimal if connected, using lifts to universal covers and height functions.

Contribution

It establishes the pivot property and entropy minimality for certain vertex shift spaces using lifts and height functions, extending understanding of their structure.

Findings

Vertex shift spaces have the pivot property.

Connected shift spaces are entropy minimal.

Proofs involve lifts to universal covers and height functions.

Abstract

Fix $d\geq 2$. Given a finite undirected graph ${\mathcal{H}}$ without self-loops and multiple edges, consider the corresponding `vertex' shift, $Hom(\mathbb{Z}^d, \mathcal{H})$ denoted by $X_{\mathcal{H}}$. In this paper we focus on $\mathcal{H}$ which is `four-cycle free'. The two main results of this paper are: $X_{\mathcal{H}}$ has the pivot property, meaning that for all distinct configurations $x,y\in X_{\mathcal{H}}$ which differ only at finitely many sites there is a sequence of configurations $x=x^1, x^2, \ldots, x^n=y\in X_{{\mathcal{H}}}$ for which the successive configurations $(x^i, x^{i+1})$ differ exactly at a single site. Further if ${\mathcal{H}}$ is connected then $X_{\mathcal{H}}$ is entropy minimal, meaning that every shift space strictly contained in $X_{\mathcal{H}}$ has strictly smaller entropy. The proofs of these seemingly disparate statements are related by the use of the `lifts' of the configurations in $X_{\mathcal{H}}$ to their universal cover and the introduction of `height functions' in this context.