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

Abstract

For an infinite word $x$, Bugeaud and Kim introduced a quantity $\mathrm{rep}(x)$ called the exponent of repetition of $x$. We prove that $\mathrm{rep}(x)=\mathrm{rep}(y)$ 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 $\mathrm{rep}(c)$ is given. This formula shows that $\mathrm{rep}(c)=\mathrm{rep}(c^{\prime })$ if $c^{\prime }$ is the characteristic Sturmian word whose slope is a quadratic irrational equivalent to $\theta$.

The results of this paper are not new