(M + 1)-step shift spaces that are not conjugate to M-step shift spaces

TL;DR

The paper confirms a conjecture that for each non-negative integer M, there exist (M+1)-step shift spaces over infinite alphabets that are not conjugate to any M-step shift spaces, expanding the understanding of shift space classifications.

Contribution

It constructs explicit examples of (M+1)-step shifts that are not conjugate to M-step shifts, proving the conjecture by Ott, Tomforde, and Willis.

Findings

Confirmed the existence of non-conjugate (M+1)-step shifts for all M

Provided a construction method for such shift spaces

Expanded the classification of shift spaces over infinite alphabets

Abstract

Recently Ott, Tomforde and Willis proposed a new approach for one sided shift spaces over infinite alphabets. In this new approach the conjugacy classes of shifts of finite type, edge shifts, and M-step shifts are distinct and the authors conjecture that for each non-negative integer M there exist an (M+1)-step shift space that is not conjugate to any M-step shift. In this short paper we build a class of (M+1)-step shifts that are not conjugate to any M-step shift and hence show that their conjecture is correct.