The exponent of repetition of the characteristic Sturmian word whose slope is a quadratic irrational

For an infinite word $x$, Bugeaud and Kim introduced a quantity $\mathrm{r}\mathrm{e}\mathrm{p}(x)$ called the exponent of repetition of $x$. We prove that $\mathrm{r}\mathrm{e}\mathrm{p}(x)=\mathrm{r}\mathrm{e}\mathrm{p}(y)$ holds for a Sturmian word $x$ and every suffix $y$ of $x$. Let $c$ be the characteristic Sturmian word of slope $\theta$. When $\theta$ is a quadratic irrational, a formula of $\mathrm{r}\mathrm{e}\mathrm{p}(c)$ is given. This formula shows that $$\mathrm{r}\mathrm{e}\mathrm{p}$© = $\mathrm{r}\mathrm{e}\mathrm{p}$(c’)$if$c’$ is the characteristic Sturmian word whose slope is a quadratic irrational equivalent to $\theta$.