Obstacles to Topological Factoring of Toeplitz shifts

TL;DR

This paper investigates the conditions under which Toeplitz shifts can be factored topologically, establishing divisibility relations between their period structures and showing Toeplitz sequences map to Toeplitz sequences under factor maps.

Contribution

It introduces the concept of constructive period structures for Toeplitz sequences and proves divisibility conditions for topological factors between Toeplitz shifts.

Findings

Topological factors imply divisibility of period structures.

Conjugacy implies equality of period structures.

Toeplitz sequences are preserved under topological factor maps.

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 $(X,\sigma,x)$ and $(Y,\sigma,y)$ 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 $ \pi:(X,\sigma)\rightarrow (Y,\sigma)$ with $\pi(x)=y$, then $q\mid p$. In particular, if $\pi$ is conjugacy, then $p=q$. We also prove that Toeplitz sequences are mapped to Toeplitz sequences through topological factorings.