  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.
