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Obstacles to Topological Factoring of Toeplitz shifts

Maryam Hosseini, Reem Yassawi

TL;DR

The paper investigates when topological factoring can occur between Toeplitz shifts. By encoding Toeplitz systems as Vershik dynamics on ERS Bratteli diagrams and exploiting constructive period structures, it shows that any topological factor sending a Toeplitz sequence with period structure $\mathbf p$ to one with $\mathbf q$ forces a divisibility relation $\mathbf q \mid \mathbf p$ (and $\mathbf p=\mathbf q$ if the factor is a conjugacy). The main argument translates factor maps into a sequence of levelwise morphisms between Bratteli diagrams and uses optimal level choices to derive necessary conditions; Toeplitz-ness is shown to be preserved under factoring. The results connect to Cobham-type statements for constant-length substitutions and emphasize the obstructions arising from the arithmetic of period structures, while also highlighting the essential role of ERS diagrams and the limits of non-ERS cases via discussed counterexamples. Overall, the work clarifies structural constraints on Toeplitz factors and provides a robust method to detect nonexistence of topological factoring in zero-dimensional systems.

Abstract

For every Toeplitz sequence $x$ with period structure $(q_i)_{i\geq 1}$, one can identify a period structure ${\bf p}=(p_i)_{i\geq 0}$ which leads to a Bratteli-Vershik realization of the associated Toeplitz shift; we refer to this period structure as {\it constructive}. Let $(X,σ,x)$ and $(Y,σ,y)$ be Toeplitz shifts where $x\in X$ and $y\in Y$ are Toeplitz sequences with constructive period structures $(p^n)_{n\geq 1}$ and $(q^n)_{n\geq 1}$, respectively. Using the Bratteli-Vershik realization of factor maps between Toeplitz shifts, we prove that if there exists a topological factoring $ π:(X,σ)\rightarrow (Y,σ)$ with $π(x)=y$, then $q\mid p$. In particular, if $π$ is conjugacy, then $p=q$. We also prove that Toeplitz sequences are mapped to Toeplitz sequences through topological factorings.

Obstacles to Topological Factoring of Toeplitz shifts

TL;DR

The paper investigates when topological factoring can occur between Toeplitz shifts. By encoding Toeplitz systems as Vershik dynamics on ERS Bratteli diagrams and exploiting constructive period structures, it shows that any topological factor sending a Toeplitz sequence with period structure to one with forces a divisibility relation (and if the factor is a conjugacy). The main argument translates factor maps into a sequence of levelwise morphisms between Bratteli diagrams and uses optimal level choices to derive necessary conditions; Toeplitz-ness is shown to be preserved under factoring. The results connect to Cobham-type statements for constant-length substitutions and emphasize the obstructions arising from the arithmetic of period structures, while also highlighting the essential role of ERS diagrams and the limits of non-ERS cases via discussed counterexamples. Overall, the work clarifies structural constraints on Toeplitz factors and provides a robust method to detect nonexistence of topological factoring in zero-dimensional systems.

Abstract

For every Toeplitz sequence with period structure , one can identify a period structure which leads to a Bratteli-Vershik realization of the associated Toeplitz shift; we refer to this period structure as {\it constructive}. Let and be Toeplitz shifts where and are Toeplitz sequences with constructive period structures and , respectively. Using the Bratteli-Vershik realization of factor maps between Toeplitz shifts, we prove that if there exists a topological factoring with , then . In particular, if is conjugacy, then . We also prove that Toeplitz sequences are mapped to Toeplitz sequences through topological factorings.

Paper Structure

This paper contains 10 sections, 12 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Prelimnaries
  3. Dynamical systems
  4. Symbolic dynamics
  5. Toeplitz shifts
  6. Substitutional dynamical systems.
  7. Kakutani-Rokhlin partitions
  8. Bratteli–Vershik realizations of zero dimensional systems
  9. Topological factoring and Bratteli-Vershik realizations
  10. Topological factoring and ERS ordered Bratteli diagrams

Key Result

Theorem 1.1

Let $(X,\sigma)$ and $(Y, \sigma)$ be two Toeplitz shifts. Suppose that $x$ and $y$ are Toeplitz sequences in $X$ and $Y$ with constructive period structures $(p^i)_{i\geq 1}$ and $(q^i)_{i\geq 1}$, respectively. If there exists a topological factor map $\pi:(X,\sigma)\rightarrow (Y,\sigma)$ with $\

Figures (3)

  • Figure 1: This figure is related to the proof of Lemma \ref{['firstlevel']}. The words linked to the finite paths are described in Example \ref{['example:one']}.
  • Figure 3: $\eta_{i+1}(w")=v'_2v'_2v'_1v'_2v'_2v'_1v'_2v'_1v'_1$, $\theta_{i+1}(v'_1)=v_1v_1v_2$, and $\theta_{i+1}(v'_2)=v_1v_2v_3$. Moreover, $\eta_{i+1}^{(e)}(w")=\eta_{i+1}^{(e')}(w")=v'_2v'_2v'_1, \eta_{i+1}^{(e")}(w")=v'_2v'_1v'_1$.
  • Figure 5: This Figure shows the counter-example mentioned in Remark \ref{['counter']}.

Theorems & Definitions (21)

  • Theorem 1.1
  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Lemma 3.1
  • proof
  • Example 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 11 more