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Asymptoticity, automorphism groups and strong orbit equivalence

Haritha Cheriyath, Sebastián Donoso

TL;DR

The paper investigates how asymptotic components and automorphism groups behave within strong orbit equivalence classes of minimal Cantor systems. It develops $oldsymbol{S}$-adic and Bratteli–Vershik techniques to realize prescribed numbers of asymptotic components (finite, countable, or continuum) inside any given class, with finite/countable cases built via primitive morphisms and recognizability, and uncountable cases tied to entropy considerations. A key contribution is proving that systems with countably many asymptotic components have zero entropy, while also producing subshifts with one asymptotic class but arbitrarily large subexponential complexity. The results yield subshifts whose automorphism groups are isomorphic to $oldsymbol{Z}$ (or trivial) within any strong orbit equivalence class and extend these constructions to Toeplitz shifts, highlighting the diversity of dynamical behavior compatible with strong orbit equivalence. Altogether, the work advances understanding of how orbit structure, entropy, complexity, and symmetry (automorphisms) can be simultaneously tuned inside a fixed orbit-equivalence class, with implications for invariant-measure structures and Toeplitz dynamics.

Abstract

Given any strong orbit equivalence class of minimal Cantor systems and any cardinal number that is finite, countable, or the continuum, we show that there exists a minimal subshift within the given class whose number of asymptotic components is exactly the given cardinal. For finite or countable ones, we explicitly construct such examples using $\mathcal{S}$-adic subshifts. We derived the uncountable case by showing that any topological dynamical system with countably many asymptotic components has zero topological entropy. We also construct systems with arbitrarily high subexponential word complexity with only one asymptotic class. We deduce that within any strong orbit equivalence class, there exists a subshift whose automorphism group is isomorphic to $\mathbb{Z}$.

Asymptoticity, automorphism groups and strong orbit equivalence

TL;DR

The paper investigates how asymptotic components and automorphism groups behave within strong orbit equivalence classes of minimal Cantor systems. It develops -adic and Bratteli–Vershik techniques to realize prescribed numbers of asymptotic components (finite, countable, or continuum) inside any given class, with finite/countable cases built via primitive morphisms and recognizability, and uncountable cases tied to entropy considerations. A key contribution is proving that systems with countably many asymptotic components have zero entropy, while also producing subshifts with one asymptotic class but arbitrarily large subexponential complexity. The results yield subshifts whose automorphism groups are isomorphic to (or trivial) within any strong orbit equivalence class and extend these constructions to Toeplitz shifts, highlighting the diversity of dynamical behavior compatible with strong orbit equivalence. Altogether, the work advances understanding of how orbit structure, entropy, complexity, and symmetry (automorphisms) can be simultaneously tuned inside a fixed orbit-equivalence class, with implications for invariant-measure structures and Toeplitz dynamics.

Abstract

Given any strong orbit equivalence class of minimal Cantor systems and any cardinal number that is finite, countable, or the continuum, we show that there exists a minimal subshift within the given class whose number of asymptotic components is exactly the given cardinal. For finite or countable ones, we explicitly construct such examples using -adic subshifts. We derived the uncountable case by showing that any topological dynamical system with countably many asymptotic components has zero topological entropy. We also construct systems with arbitrarily high subexponential word complexity with only one asymptotic class. We deduce that within any strong orbit equivalence class, there exists a subshift whose automorphism group is isomorphic to .

Paper Structure

This paper contains 29 sections, 34 theorems, 24 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization of the paper
  4. Preliminaries
  5. Minimal Cantor systems
  6. Symbolic dynamical systems
  7. $\mathcal{S}$-adic subshifts
  8. Recognizability of morphisms
  9. Toeplitz subshifts
  10. Bratteli-Vershik systems
  11. Bratteli diagrams
  12. Ordered Bratteli diagrams and Vershik map
  13. Strong orbit equivalence
  14. Morphisms read on an ordered Bratteli diagram
  15. Asymptotic components and automorphism groups
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $(X,T)$ be a topological dynamical system with at most countably many asymptotic components. Then $(X,T)$ has topological entropy zero.

Figures (8)

  • Figure 1: Cutting points of $\tau(x)$: $C_{\tau}(x)=\{c_i\mid\ i\in{\mathbb Z}\}$
  • Figure 2: Bifurcation point and the signal
  • Figure 3: Construction of cylinders to get $x^i$
  • Figure 4: Construction of cylinders to get $y^i$
  • Figure 5: Factorization of $x$ and $y$ at level $n$
  • ...and 3 more figures

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof : Proof of Theorem \ref{['thm:main3']}
  • Corollary 3.1
  • ...and 58 more