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Strictly ergodic Toeplitz $\mathbb{Z}^d$-subshifts with arbitrary entropy

Jamal Drewlo

Abstract

In this work, we present a comprehensive construction that proves the existence of strictly ergodic Toeplitz $\mathbb{Z}^d$-subshifts which admit arbitrary given entropy. Moreover, any of these constructed subshifts will have the same maximal equicontinuous factor.

Strictly ergodic Toeplitz $\mathbb{Z}^d$-subshifts with arbitrary entropy

Abstract

In this work, we present a comprehensive construction that proves the existence of strictly ergodic Toeplitz -subshifts which admit arbitrary given entropy. Moreover, any of these constructed subshifts will have the same maximal equicontinuous factor.

Paper Structure

This paper contains 14 sections, 30 theorems, 115 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notions
  4. Subshifts
  5. Toeplitz arrays and Toeplitz subshifts
  6. $G$-odometers
  7. Toeplitz subshifts
  8. Decomposing a residually finite, abelian group
  9. Unique ergodicity and entropy for Toeplitz subshifts
  10. The construction
  11. Obtaining a fundamental domain
  12. Three sequences
  13. Proof of Theorem A
  14. Consequences

Key Result

Theorem 1

Let $k\geq 2$ and consider the alphabet $\Sigma_k = \left\{ 0,\ldots,k-1 \right\}$. Then, for every $0<h<\log k$ there exists a Toeplitz-$\mathbb{Z}^d$-array $x\in (\Sigma_k)^{\mathbb{Z}^d}$ such that the Toeplitz subshift $(\overline{O_{\mathbb{Z}^d}}(x),\mathbb{Z}^d)$ is strictly ergodic and $h_{\

Figures (3)

  • Figure 1: A visualization of Example \ref{['ex: theta maps']}. Shifted copies of the fundamental domains are marked by dashed borders.
  • Figure 2: The decomposition of $D_n+g$ into $\biguplus_{\gamma\in I_{t,n}(g)} (D_t+\gamma)$ and $R_{t,n}(g)$. The green dots are the elements of the subgroup $\Gamma_t$ and the brown crosses are the points in the set $I_{t,n}(g)$.
  • Figure 3: A visualization of the decomposition in \ref{['eq: D_t D_s D_n']}. In this particular setting, the set $I_{s,n}(g)$ consists of a single point marked by the pink star. The dark green, dashed area corresponds to the double union in \ref{['eq: D_t D_s D_n']}, whereas the light green, dashed squares are the sets $D_t+\gamma \subseteq R_{s,n}(g)\:(\gamma\in \Gamma_t)$.

Theorems & Definitions (53)

  • Theorem : Theorem A
  • Theorem 2.1
  • Remark 2.2
  • Lemma 3.1: CortezPetite2008Odometers
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • proof
  • Proposition 3.5: CortezPetite2008Odometers
  • ...and 43 more