Four-cycle free graphs, height functions, the pivot property and entropy minimality. (8th March 2016) - CHANDGOTIA, NISHANT
Abstract : Fix $d≥ 2$ . Given a finite undirected graph ${\mathcal{H}}$ without self-loops and multiple edges, consider the corresponding 'vertex' shift, $\text{Hom}(\mathbb{Z}^{d}, {\mathcal{H}})$, denoted by $X_{{\mathcal{H}}}$ . In this paper, we focus on ${\mathcal{H}}$ which is 'four-cycle free'. There are two main results of this paper. Firstly, that $X_{{\mathcal{H}}}$ has the pivot property, meaning that, for all distinct configurations $x, y\in X_{{\mathcal{H}}}$, which differ only at a finite number of sites, there is a sequence of configurations $x=x^{1}, x^{2}, …, x^{n}=y\in X_{{\mathcal{H}}}$ for which the successive configurations $x^{i}, x^{i+1}$ differ exactly at a single site. Secondly, 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 the universal cover of ${\mathcal{H}}$ and the introduction of 'height functions' in this context.