Mixing Properties for Hom-Shifts and the Distance between Walks on Associated Graphs

Abstract

Let $\mathcal{H}$ be a finite connected undirected graph and ${\mathcal{H}}_{walk}$ be the graph of bi-infinite walks on $\mathcal{H}$; two such walks $\{x_i{\}}_{i\in \mathbb{Z}}$ and $\{y_i{\}}_{i\in \mathbb{Z}}$ are said to be adjacent if $x_i$ is adjacent to $y_i$ for all $i\in \mathbb{Z}$. We consider the question: Given a graph $\mathcal{H}$ when is the diameter (with respect to the graph metric) of ${\mathcal{H}}_{walk}$ 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 ${\mathbb{Z}}^d$ with respect to the standard generators to $\mathcal{H}$) and are the subject of this paper.

Corrected typo in the title on the Arxiv page

References in corpus (1)

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

Cited by in corpus (3)

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• Stem and topological entropy on Cayley trees
• Decidability of irreducible tree shifts of finite type