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

TL;DR

This paper investigates the repetition exponent of characteristic Sturmian words, providing a formula for quadratic irrational slopes and establishing invariance of this exponent among equivalent slopes.

Contribution

It derives a formula for the repetition exponent of characteristic Sturmian words with quadratic irrational slopes and shows its invariance under slope equivalence.

Findings

The repetition exponent is equal for all suffixes of a Sturmian word.

A specific formula for the exponent when the slope is quadratic irrational.

The exponent remains constant for equivalent quadratic irrational slopes.

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 $\mathrm{rep}(c) = \mathrm{rep}(c')$ if $c'$ is the characteristic Sturmian word whose slope is a quadratic irrational equivalent to $\theta$.