Two-sided shift spaces over infinite alphabets

TL;DR

This paper introduces a new class of two-sided shift spaces over infinite alphabets, extending classical symbolic dynamics results and providing positive answers to open problems in the field.

Contribution

It develops a notion of two-sided shift spaces over infinite alphabets with continuous shift maps, generalizing prior one-sided models and classical results.

Findings

Two-sided shift spaces are compact Hausdorff spaces with continuous shift maps.

Many classical symbolic dynamics results hold for these two-sided shift spaces.

The paper proves that any M-step shift is conjugate to an edge shift space in the two-sided setting.

Abstract

Ott, Tomforde, and Willis proposed a useful compactification for one-sided shifts over infinite alphabets. Building from their idea we develop a notion of two-sided shift spaces over infinite alphabets, with an eye towards generalizing a result of Kitchens. As with the one-sided shifts over infinite alphabets our shift spaces are compact Hausdorff spaces but, in contrast to the one-sided setting, our shift map is continuous everywhere. We show that many of the classical results from symbolic dynamics are still true for our two-sided shift spaces. In particular, while for one-sided shifts the problem about whether or not any $M$-step shift is conjugate to an edge shift space is open, for two-sided shifts we can give a positive answer for this question.