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

Takao Watanabe

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arXiv · 2201.06689

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

Takao Watanabe

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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 $\theta$.

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Takao Watanabe. 2022-01-18. The exponent of repetition of the characteristic Sturmian word whose slope is a quadratic irrational. https://arxiv.org/abs/2201.06689

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