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

DOI: 10.2140/pjm.2018.294.41

Abstract

Let $\mathcal{\mathscr{H}}$ be a finite connected undirected graph and ${\mathcal{\mathscr{H}}}_{\mathrm{\text{ walk}}}^2$ be the graph of bi-infinite walks on $\mathcal{\mathscr{H}}$; two such walks ${\left\{x_i\right\}}_{i\in \mathbb{Z}}$ and ${\left\{y_i\right\}}_{i\in \mathbb{Z}}$ 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 $\mathcal{\mathscr{H}}$, when is the diameter (with respect to the graph metric) of ${\mathcal{\mathscr{H}}}_{\mathrm{\text{ walk}}}^2$ 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 $\mathcal{\mathscr{H}}$) 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

Milestones

Received: 5 October 2016

Revised: 28 May 2017

Accepted: 19 October 2017

Published: 5 January 2018