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

Collin Bleak; Feyishayo Olukoya

doi:10.3934/dcds.2025142

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

Collin Bleak^*
, Feyishayo Olukoya

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

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 exist an element ψ in the group of automorphisms of the one-sided shift Aut({0, 1, . . . , n − 1}^ℕ, σ_n) so that all points of {0, 1, . . . , n − 1}^ℕ have orbits of length n under ψ and ψ is not conjugate to a permutation?

Here, by a permutation we mean an automorphism of the one-sided shift dynamical system induced by a permutation of the symbol set {0, 1, . . . , n − 1}.

We resolve this question by showing constructively that any ψ 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).

Original language	English
Pages (from-to)	272-311
Number of pages	40
Journal	Discrete and Continuous Dynamical Systems
Volume	48
Early online date	1 Sept 2025
DOIs	https://doi.org/10.3934/dcds.2025142
Publication status	Published - 1 Mar 2026

Keywords

Group theory
Dynamics
Automorphisms of the one-sided shift
Conjugacy
Transducers
Aynchronizing automata

Access to Document

10.3934/dcds.2025142

Bleak_2025_DCDS_Conjugacy-automorphisms-transducers_AAM_CCBYNCND
Copyright © 2025 the Authors. This work has been made available online in accordance with the University of St Andrews Open Access policy. This accepted manuscript is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License, which permits use, distribution, and reproduction in any medium, in unadapted form only, for non commercial purposes only, provided the original author and source are credited. The final published version of this work is available at https://doi.org/10.3934/dcds.2025142
Accepted author manuscript, 572 KBLicence: CC BY-NC-ND

Handle.net

Persistent link to portal record

1 Finished
1 Active

The Higman Thompson groups: The Higman-Thompson groups, their generalisations, and automorphisms of shift spaces
Olukoya, S. (PI)
EPSRC
1/02/24 → 31/01/27
Project: Fellowship
Bi-synchronizing automata: Bi-synchronizing automata, outer automorphism groups of Higman-Thompson groups, and automorphisms of the shift
Bleak, C. (PI) & Cameron, P. (CoI)
1/05/18 → 30/04/21
Project: Standard

1 Preprint

Conjugacy for certain automorphisms of the one-sided shift via transducers
Bleak, C. & Olukoya, F., 31 Jan 2023, arXiv, 53 p.
Research output: Working paper › Preprint

Open Access

Cite this

@article{004cd140cfa8405a9d49aaf89cf0f53f,

title = "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 exist an element ψ in the group of automorphisms of the one-sided shift Aut(\{0, 1, . . . , n − 1\}ℕ, σn) so that all points of \{0, 1, . . . , n − 1\}ℕ have orbits of length n under ψ and ψ is not conjugate to a permutation? Here, by a permutation we mean an automorphism of the one-sided shift dynamical system induced by a permutation of the symbol set \{0, 1, . . . , n − 1\}. We resolve this question by showing constructively that any ψ 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).",

keywords = "Group theory, Dynamics, Automorphisms of the one-sided shift, Conjugacy, Transducers, Aynchronizing automata",

author = "Collin Bleak and Feyishayo Olukoya",

note = "Funding: The authors are grateful for partial support from EPSRC research grants EP/R032866/1. The second author is additionally grateful for support from Leverhulme Trust Research Project Grant RPG-2017-159, LMS ECF grant ECF-1920-105 and EPSRC research grant EP/X02606X/1.",

year = "2026",

month = mar,

day = "1",

doi = "10.3934/dcds.2025142",

language = "English",

volume = "48",

pages = "272--311",

journal = "Discrete and Continuous Dynamical Systems",

issn = "1078-0947",

publisher = "American Institute of Mathematical Sciences",

}

TY - JOUR

T1 - Conjugacy for certain automorphisms of the one-sided shift via transducers

AU - Bleak, Collin

AU - Olukoya, Feyishayo

N1 - Funding: The authors are grateful for partial support from EPSRC research grants EP/R032866/1. The second author is additionally grateful for support from Leverhulme Trust Research Project Grant RPG-2017-159, LMS ECF grant ECF-1920-105 and EPSRC research grant EP/X02606X/1.

PY - 2026/3/1

Y1 - 2026/3/1

N2 - 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 exist an element ψ in the group of automorphisms of the one-sided shift Aut({0, 1, . . . , n − 1}ℕ, σn) so that all points of {0, 1, . . . , n − 1}ℕ have orbits of length n under ψ and ψ is not conjugate to a permutation? Here, by a permutation we mean an automorphism of the one-sided shift dynamical system induced by a permutation of the symbol set {0, 1, . . . , n − 1}. We resolve this question by showing constructively that any ψ 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).

AB - 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 exist an element ψ in the group of automorphisms of the one-sided shift Aut({0, 1, . . . , n − 1}ℕ, σn) so that all points of {0, 1, . . . , n − 1}ℕ have orbits of length n under ψ and ψ is not conjugate to a permutation? Here, by a permutation we mean an automorphism of the one-sided shift dynamical system induced by a permutation of the symbol set {0, 1, . . . , n − 1}. We resolve this question by showing constructively that any ψ 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).

KW - Group theory

KW - Dynamics

KW - Automorphisms of the one-sided shift

KW - Conjugacy

KW - Transducers

KW - Aynchronizing automata

U2 - 10.3934/dcds.2025142

DO - 10.3934/dcds.2025142

M3 - Article

SN - 1078-0947

VL - 48

SP - 272

EP - 311

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

ER -

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

Abstract

Keywords

Access to Document

Handle.net

Fingerprint

Projects

The Higman Thompson groups: The Higman-Thompson groups, their generalisations, and automorphisms of shift spaces

Bi-synchronizing automata: Bi-synchronizing automata, outer automorphism groups of Higman-Thompson groups, and automorphisms of the shift

Research output

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

Cite this