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

Fix . Given a finite undirected graph without self-loops and multiple edges, consider the corresponding ‘vertex’ shift, , denoted by . In this paper, we focus on which is ‘four-cycle free’. There are two main results of this paper. Firstly, that has the pivot property, meaning that, for all distinct configurations , which differ only at a finite number of sites, there is a sequence of configurations for which the successive configurations differ exactly at a single site. Secondly, if is connected ,then is entropy minimal, meaning that every shift space strictly contained in has strictly smaller entropy. The proofs of these seemingly disparate statements are related by the use of the ‘lifts’ of the configurations in to the universal cover of and the introduction of ‘height functions’ in this context.