On non-repetitive complexity of Arnoux-Rauzy words

Abstract

The non-repetitive complexity $nr{\mathcal{C}}_{\mathbf{u}}$ and the initial non-repetitive complexity $inr{\mathcal{C}}_{\mathbf{u}}$ are functions which reflect the structure of the infinite word $\mathbf{u}$ with respect to the repetitions of factors of a given length. We determine $nr{\mathcal{C}}_{\mathbf{u}}$ for the Arnoux-Rauzy words and $inr{\mathcal{C}}_{\mathbf{u}}$ 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 $nr{\mathcal{C}}_{\mathbf{u}}$ and $inr{\mathcal{C}}_{\mathbf{u}}$ for the $d$-bonacci word.