Conjugacy for certain automorphisms of the one-sided shift via transducers
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).
Citation
Bleak , C & Olukoya , F 2023 ' Conjugacy for certain automorphisms of the one-sided shift via transducers ' arXiv . https://doi.org/10.48550/arXiv.2301.13570
Type
Working or discussion paper
Rights
(c) 2023 the authors. This open access preprint is available under a Creative Commons Attribution licence
Description
53 Pages, 12 figuresCollections
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