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arXiv · 2002.12593

On non-repetitive complexity of Arnoux-Rauzy words

Kateřina Medková·Edita Pelantová·Élise Vandomme
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Abstract

The non-repetitive complexity $nr\mathcal{C}_{\bf u}$ and the initial non-repetitive complexity $inr\mathcal{C}_{\bf u}$ are functions which reflect the structure of the infinite word ${\bf u}$ with respect to the repetitions of factors of a given length. We determine $nr\mathcal{C}_{\bf u}$ for the Arnoux-Rauzy words and $inr\mathcal{C}_{\bf u}$ for the standard Arnoux-Rauzy words. Our main tools are $S$-adic representation of Arnoux-Rauzy words and description of return words to their factors. The formulas we obtain are then used to evaluate $nr\mathcal{C}_{\bf u}$ and $inr\mathcal{C}_{\bf u}$ for the $d$-bonacci word.

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Kateřina Medková, Edita Pelantová, Élise Vandomme. 2020-02-28. On non-repetitive complexity of Arnoux-Rauzy words. https://arxiv.org/abs/2002.12593

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