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

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.