Vol. 294, No. 1, 2018

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Mixing properties for hom-shifts and the distance between walks on associated graphs

Nishant Chandgotia and Brian Marcus

Vol. 294 (2018), No. 1, 41–69
Abstract

Let $\mathsc{ℋ}$ be a finite connected undirected graph and be the graph of bi-infinite walks on $\mathsc{ℋ}$; two such walks ${\left\{{x}_{i}\right\}}_{i\in ℤ}$ and ${\left\{{y}_{i}\right\}}_{i\in ℤ}$ are said to be adjacent if ${x}_{i}$ is adjacent to ${y}_{i}$ for all $i\in ℤ$. We consider the question: Given a graph $\mathsc{ℋ}$, when is the diameter (with respect to the graph metric) of finite? Such questions arise while studying mixing properties of hom-shifts (shift spaces which arise as the space of graph homomorphisms from the Cayley graph of ${ℤ}^{d}$ with respect to the standard generators to $\mathsc{ℋ}$) and are the subject of this paper.

Keywords
walks on graphs, folding, block-gluing, symbolic dynamics, strong irreducibility, universal covers
Mathematical Subject Classification 2010
Primary: 37B10
Secondary: 68R10, 82B20