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 H without self-loops and multiple edges, consider the corresponding 'vertex' shift, Hom(Z^d, H), denoted by X_H . In this paper, we focus on H which is 'four-cycle free'. There are two main results of this paper. Firstly, that X_H has the pivot property, meaning that, for all distinct configurations x, yX_H, which differ only at a finite number of sites, there is a sequence of configurations x=x^1, x^2, …, x^n=yX_H for which the successive configurations x^i, x^i+1 differ exactly at a single site. Secondly, if H is connected, then X_H is entropy minimal, meaning that every shift space strictly contained in X_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_H to the universal cover of H and the introduction of 'height functions' in this context.