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

Collin Bleak, Feyishayo Olukoya

TL;DR

The paper resolves a long-standing open problem by showing that any finite-order automorphism ψ of the one-sided $n$-shift, with all points moving on orbits of length $n$, is conjugate to a permutation, and more generally to a product of disjoint cycles of length $N$ for any divisor $N$ of $n$. It develops a constructive framework based on transducers, strongly synchronizing automata, and folded de Bruijn graphs, enabling a step-by-step reduction to minimal supports and explicit conjugacies. The authors introduce the dual transducer, minimal digraphs, and a regime of relabellings, shadow states, and rebraiding to systematically shrink conjugacy class representatives while preserving the automorphism. The approach not only resolves the posed question but also yields a robust toolkit for analyzing finite-order automorphisms in this setting, with potential extensions to broader $n$ and implicit connections to automorphism groups of subshifts of finite type.

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 $ψ$ in the group of automorphisms of the one-sided shift $\operatorname{Aut}(\{0,1,\ldots,n-1\}^{\mathbb{N}}, σ_{n})$ so that all points of $\{0,1,\ldots,n-1\}^{\mathbb{N}}$ have orbits of length $n$ under $ψ$ and $ψ$ is not conjugate to a permutation?" Here, by a 'permutation' we mean an automorphism of 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 $ψ$ 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).

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

TL;DR

The paper resolves a long-standing open problem by showing that any finite-order automorphism ψ of the one-sided -shift, with all points moving on orbits of length , is conjugate to a permutation, and more generally to a product of disjoint cycles of length for any divisor of . It develops a constructive framework based on transducers, strongly synchronizing automata, and folded de Bruijn graphs, enabling a step-by-step reduction to minimal supports and explicit conjugacies. The authors introduce the dual transducer, minimal digraphs, and a regime of relabellings, shadow states, and rebraiding to systematically shrink conjugacy class representatives while preserving the automorphism. The approach not only resolves the posed question but also yields a robust toolkit for analyzing finite-order automorphisms in this setting, with potential extensions to broader and implicit connections to automorphism groups of subshifts of finite type.

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 in the group of automorphisms of the one-sided shift so that all points of have orbits of length under and 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 . We resolve this question by showing that any 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).
Paper Structure (33 sections, 26 theorems, 38 equations, 9 figures, 1 table)

This paper contains 33 sections, 26 theorems, 38 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The natural numbers and some of its subsets
  4. Words and infinite sequences
  5. One-sided shift space
  6. Automata and transducers
  7. Automata
  8. Transducers
  9. Transducer toolkit
  10. Automorphisms of the one-sided shift and synchronizing automata
  11. Synchronizing automata
  12. Automorphisms of the one-sided shift from synchronizing transducers
  13. Finite order elements of Lg as automorphisms of folded de Bruijn graphs
  14. de Bruijn graphs, foldings and synchronizing automata
  15. Finite order elements of Lg as automorphisms of synchronizing automata
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $n\geq 2$ be an integer and suppose $\phi\in \hbox{Aut}({X_{n}^{\mathbb{N}}, \sigma_{n}})$ is a finite order element. There is an effective process for determining $\Gamma_\phi$, the minimal folded de Bruijn graph on an $n$ letter alphabet, so that $\phi$ induces a natural automorphism of $\Gamm

Figures (9)

  • Figure 1: A transducer over $X_2$
  • Figure 2: The de Bruijn graph $G(3,2)$.
  • Figure 3: An element $H \in \mathcal{H}_{6}$ of order $6$.
  • Figure 4: The level 2 dual of $H$.
  • Figure 5: The automaton $\mathscr{A}({H}^{\vee}_{2})$
  • ...and 4 more figures

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Example 3.4
  • Lemma 3.5
  • ...and 45 more