Conjugacy for certain automorphisms of the one-sided shift via transducers

TL;DR

This paper proves that any automorphism of a one-sided shift with all points having orbits of length n must be conjugate to a permutation, resolving an open problem from 1990 using automata theory.

Contribution

It demonstrates that such automorphisms are necessarily conjugate to permutations, extending automata techniques to solve a longstanding open problem.

Findings

Any automorphism with all points of orbit length n is conjugate to a permutation.

Uses strongly synchronizing automata to analyze automorphism structure.

Provides a largely self-contained proof extending previous automata methods.

Abstract

We address the following open problem, implicit in the 1990 article "Automorphisms of one-sided subshifts of finite type" of Boyle, Franks and Kitchens (BFK): "Does there exists an element $\psi$ in the group of automorphisms of the one-sided shift $\operatorname{Aut}(\{0,1,\ldots,n-1\}^{\mathbb{N}}, \sigma_{n})$ so that all points of $\{0,1,\ldots,n-1\}^{\mathbb{N}}$ have orbits of length $n$ under $\psi$ and $\psi$ is not conjugate to a permutation?" Here, by a 'permutation' we mean an automorphism of one-sided shift dynamical system induced by a permutation of the symbol set $\{0,1,\ldots,n-1\}$. We resolve this question by showing that any $\psi$ with properties as above must be conjugate to a permutation. Our techniques naturally extend those of BFK using the strongly synchronizing automata technology developed here and in several articles of the authors and collaborators (although, this article has been written to be largely self-contained).