Tame or wild Toeplitz shifts

TL;DR

This paper studies the tameness of Toeplitz shifts, introducing extended diagrams and characterizing tameness via the countability of singular fiber orbits, with applications to substitution subshifts.

Contribution

It introduces extended Bratteli-Vershik diagrams and provides a precise criterion for tameness in Toeplitz shifts based on orbit countability.

Findings

Tame Toeplitz shifts with finite rank are characterized by countably many singular fiber orbits.

The paper establishes an if-and-only-if condition for tameness in terms of orbit count.

Examples demonstrate the necessity of the assumptions in the main results.

Abstract

We investigate tameness of Toeplitz shifts. By introducing the notion of extended Bratteli-Vershik diagrams, we show that such shifts with finite Toeplitz rank are tame if and only if there are at most countably many orbits of singular fibres over the maximal equicontinuous factor. The ideas are illustrated using the class of substitution subshifts. A body of elaborate examples shows that the assumptions of our results cannot be relaxed.