On non-repetitive complexity of Arnoux-Rauzy words

Abstract

The non-repetitive complexity nrC u and the initial non-repetitive complexity inrC u are functions which reflect the structure of the infinite word u with respect to the repetitions of factors of a given length. We determine nrC u for the Arnoux-Rauzy words and inrC 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 nrC u and inrC u for the d-bonacci word.