The spectrum of the exponents of repetition

TL;DR

This paper investigates the spectrum of the exponent of repetition for Sturmian words, identifying bounds, extremal values, and the structure of the set of these exponents for specific slopes like the golden ratio.

Contribution

It determines the minimum and maximum of the exponent of repetition for Sturmian words with certain slopes and analyzes the topological structure of the set of these exponents.

Findings

Minimum exponent for bounded partial quotients

Maximum and minimum of the set for the golden ratio

Isolation and limit points among the largest exponents

Abstract

For an infinite word $\mathbf{x}$, Bugeaud and Kim introduced a new complexity function $\text{rep}(\mathbf{x})$ which is called the exponent of repetition of $\mathbf{x}$. They showed $1\le \text{rep}(\mathbf{x}) \le \sqrt{10}-\frac{3}{2}$ for any Sturmian word $\mathbf{x}$. Ohnaka and Watanabe found a gap in the set of the exponents of repetition of Sturmian words. For an irrational number $\theta\in(0,1)$, let \[ \mathscr{L}(\theta):=\{\text{rep}(\mathbf{x}):\textrm{$\mathbf{x}$ is an Sturmian word of slope $\theta$}\}.\] In this article, we look into $\mathscr{L}(\theta)$. The minimum of $\mathscr{L}(\theta)$ is determined where $\theta$ has bounded partial quotients in its continued fraction expression. In particular, we find out the maximum and the minimum of $\mathscr{L}(\varphi)$ where $\varphi:=\frac{\sqrt{5}-1}{2}$ is the fraction part of the golden ratio. Furthermore, we show that the three largest values are isolated points in $\mathscr{L}(\varphi)$ and the fourth largest point is a limit point of $\mathscr{L}(\varphi)$.