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Tame or wild Toeplitz shifts

Gabriel Fuhrmann, Johannes Kellendonk, Reem Yassawi

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.

Tame or wild Toeplitz shifts

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.

Paper Structure

This paper contains 22 sections, 42 theorems, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Basic notions and notation
  3. Toeplitz shifts
  4. Odometers and Toeplitz shifts
  5. Constant length substitutions
  6. The Bratteli-Vershik representation of a Toeplitz shift
  7. Bratteli-Vershik systems
  8. Toeplitz Bratteli-Vershik systems
  9. Shift interpretation
  10. Toeplitz Bratteli Vershik diagrams for constant length substitutions
  11. The extended Bratteli diagram
  12. Thick Toeplitz shifts
  13. Toeplitz systems and non-tameness
  14. Almost automorphy and semicocycle extensions
  15. Criteria for tameness of almost automorphic shifts
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $(X, T)$ be a Toeplitz shift of finite Toeplitz rank. Then $(X,T)$ is tame if and only if its maximal equicontinuous factor has only countably many singular points.

Figures (3)

  • Figure 2.1: The graph $\mathcal{G}_\theta$ for the substitutions from Example \ref{['ex:examplezero']} (left) and Example \ref{['ex:examplezerob']} (right). We used blue dotted and violet dashed lines for better comparison with Figure \ref{['thickness-not-rank-picture-1']}.
  • Figure 2.2: One level of the stationary extended Bratteli diagram of Example \ref{['ex:first']}. The order is indicated through colour: black, blue dotted, violet dashed and red edges correspond to order label 0,1,2 and 3 respectively. The grey vertices are not extendable. Red edges are finer than black edges if viewed without colour.
  • Figure 2.3: On the right, we see the first levels of the extended Bratteli diagram of the one-sided shifts for Example \ref{['exp:oxtoby']} with $\ell_1=3$ and $\ell_2=5$. The order is indicated through colour: black, blue dotted, violet dashed, green densely dotted, and red edges correspond to order label $0,1,2,3$ and $4$ respectively. Red edges are finer than black edges if viewed without colour. As more levels are added, there are increasingly many edges between vertices labelled $\{a,b\}$ in consecutive levels. This is to be contrasted with the one-sided period-doubling substitution shift (on the left), where $\ell_n=2$ for all $n$, and which is tame (again, black and blue dotted edges correspond to order label $0$ and $1$, respectively). It has thickness one.

Theorems & Definitions (87)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5: cf. KerrLi2007 & FuhrmannKwietniak2020
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Example A
  • ...and 77 more