Ultragraphs and shifts spaces over infinite alphabets

TL;DR

This paper advances the understanding of shift spaces over infinite alphabets by characterizing them via ultragraphs, exploring shifts of finite type, and connecting ultragraph C*-algebras with partial crossed products.

Contribution

It introduces ultragraphs as a framework for one-sided shift spaces over infinite alphabets and addresses conjectures about shifts of finite type and their conjugacy properties.

Findings

Existence of shifts of finite type not conjugate to edge shifts of graphs.

Edge shifts of ultragraphs can be shifts of finite type but not conjugate to full shifts.

Ultragraph C*-algebras can be realized as partial crossed products.

Abstract

In this paper we further develop the theory of one sided shift spaces over infinite alphabets, characterizing one-step shifts as edge shifts of ultragraphs and partially answering a conjecture regarding shifts of finite type (we show that there exists shifts of finite type that are not conjugate, via a conjugacy that is eventually finite periodic, to an edge shift of a graph ). We also show that there exists edge shifts of ultragraphs that are shifts of finite type, but are not conjugate to a full shift, a result that is not true for edge shifts of graphs. One of the key results needed in the proofs of our conclusions is the realization of a class of ultragraph C*-algebras as partial crossed products, a result of interest on its own.