The spectrum of the exponents of repetition

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}$foranySturmianword$$\mathbf{x}$$. 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 $$\mathcal{L}(\theta ):=\{\text{rep}(\mathbf{x}):\mathbf{x}\text{ is an Sturmian word of slope }\theta \}\mathrm{.}$$ 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 $\mathcal{L}(\phi )$.