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 \[ \mathscr{L}(θ):=\{\text{rep}(\mathbf{x}):\textrm{$\mathbf{x}$ is an Sturmian word of slope $\theta$}\}.\] In this article, we look into $\mathcal{L}(\theta )$. The minimum of $\mathcal{L}(\theta )$ is determined where $\theta$ has bounded partial quotients in its continued fraction expression. In particular, we find out the maximum and the minimum of $\mathcal{L}(\phi )$ where $\phi :=\frac{\sqrt{5}-1}{2}$ is the fraction part of the golden ratio. Furthermore, we show that the three largest values are isolated points in $\mathcal{L}(\phi )$ and the fourth largest point is a limit point of $\mathcal{L}(\phi )$.