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.
Categories: math.DS