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(S,w)-Gap Shifts and Their Entropy

Cristian Ramirez, Amy Somers

Abstract

The $S$-gap shifts have a dynamically and combinatorially rich structure. Dynamical properties of the $S$-gap shift can be related to the properties of the set $S$. This interplay is particularly interesting when $S$ is not syndetic such as when $S$ is the set of prime numbers or when $S=\{2^n\}$. It is a well known result that the entropy of the $S$-gap shift is given by $h(X) = \log λ$, where $λ>0$ is the unique solution to the equation $\sum_{n \in S} λ^{-(n+1)}=1$. Fix a point $w$ of the full shift $\{1,2, \dots, k\}^\mathbb{Z}$. We introduce the $(S,w)$-gap shift which is a generalization of the $S$-gap shift consisting of sequences in $\{0,1, \dots, k\}^\mathbb{Z}$ in which any two $0$'s are separated by a word $u$ appearing in $w$ such that $|u|\in S$. We extend the formula for the entropy of the $S$-gap shift to a formula describing the entropy of this new class of shift spaces. Additionally we investigate the dynamical properties including irreducibility and mixing of this generalization of the $S$-gap shift.

(S,w)-Gap Shifts and Their Entropy

Abstract

The -gap shifts have a dynamically and combinatorially rich structure. Dynamical properties of the -gap shift can be related to the properties of the set . This interplay is particularly interesting when is not syndetic such as when is the set of prime numbers or when . It is a well known result that the entropy of the -gap shift is given by , where is the unique solution to the equation . Fix a point of the full shift . We introduce the -gap shift which is a generalization of the -gap shift consisting of sequences in in which any two 's are separated by a word appearing in such that . We extend the formula for the entropy of the -gap shift to a formula describing the entropy of this new class of shift spaces. Additionally we investigate the dynamical properties including irreducibility and mixing of this generalization of the -gap shift.
Paper Structure (5 sections, 11 theorems, 58 equations)

This paper contains 5 sections, 11 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Background
  3. $S$-Gap Shifts
  4. $(S,w)$-gap Shifts
  5. Entropy

Key Result

Theorem 1.1

Let $w \in \{1,2, \dots, k\}^\mathbb{Z}$ and $S \subseteq \mathbb{Z}_{\geq 0}$. Then $h(X_w(S))=\log \lambda$ where $\lambda>0$ is the unique positive solution to

Theorems & Definitions (35)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 25 more