Four-cycle free graphs, height functions, the pivot property and entropy minimality

作者：Chandgotia Nishant^*

来源：Ergodic Theory and Dynamical Systems, 2017, 37(04): 1102-1132.

DOI：10.1017/etds.2015.88

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, y is an element of X-H, which differ only at a finite number of sites, there is a sequence of configurations x = x(1), x(2), . . . , x(n) = y is an element of X-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.

• 出版日期2017-6