Dynamics and Topology of S-gap Shifts

TL;DR

This paper characterizes the dynamical properties of S-gap shifts, showing conditions for finite type, almost-finite type, and sofic shifts, and establishes a correspondence with a set of real numbers to analyze their topology and measure.

Contribution

It provides a complete classification of S-gap shifts based on the properties of the sequence S and introduces a novel correspondence with real numbers to study their structure.

Findings

SFT if and only if S is finite or cofinite

AFT if and only if the difference sequence is eventually constant

Sofic if and only if the difference sequence is eventually periodic

Abstract

Let $S=\{s_i\in\mathbb N\cup\{0\}:0\leq s_i<s_{i+1}\}$ and let $d_{0}=s_{0}$ and $\Delta(S)=\{d_{n}\}_{n}$ where $d_{n}=s_{n}-s_{n-1}$. In this note, we show that an $S$-gap shift is subshift of finite type (SFT) if and only if $S$ is finite or cofinite, is almost-finite-type (AFT) if and only if $\Delta(S)$ is eventually constant and is sofic if and only if $\Delta(S)$ is eventually periodic. We also show that there is a one-to-one correspondence between the set of all $S$-gap shifts and $\{r \in \mathbb R: r \geq 0\}\backslash \{\frac{1}{n}: n \in {\mathbb N}\}$ up to conjugacy. This enables us to induce a topology and measure structure on the set of all $S$-gaps. By using this, we give the frequency of certain $S$-gap shifts with respect to their dynamical properties.