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

Fix $d\ge 2$. 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}$}}$forwhichthesuccessiveconfigurations$(x^i, x^{i+1})$differexactlyatasinglesite\mathrm{.}Furtherif${$\mathcal{H}$}$isconnectedthen$X_{$\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.