The spacetime of a shift endomorphism

TL;DR

This paper explores the structure of automorphisms and endomorphisms of one-dimensional shift spaces by examining their associated two-dimensional shift systems, revealing links between nonexpansive subspaces and dynamical properties.

Contribution

It introduces a novel perspective by analyzing the two-dimensional shift system associated with a single automorphism, connecting geometric properties to dynamical behavior.

Findings

Automorphism groups vary significantly between positive and zero entropy shifts.

Nonexpansive subspaces relate to specific dynamical properties of automorphisms.

The approach provides new insights into the structure of shift automorphisms.

Abstract

The automorphism group of a one dimensional shift space over a finite alphabet exhibits different types of behavior: for a large class with positive entropy, it contains a rich collection of subgroups, while for many shifts of zero entropy, there are strong constraints on the automorphism group. We view this from a different perspective, considering a single automorphism (and sometimes endomorphism) and studying the naturally associated two dimensional shift system. In particular, we describe the relation between nonexpansive subspaces in this two dimensional system and dynamical properties of an automorphism of the shift.