The spectrum of the exponents of repetition

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 $\mathscr{L}(\varphi)$.