Frequencies of factors in Arnoux–Rauzy sequences

Abstract

V. Berthe showed that the frequencies of factors in a Sturmian word of slope α, as well as the number of factors with a given frequency, can be expressed in terms of the continued fraction expansion of α. In this paper we describe a multi-dimensional continued fraction process associated with a class of sequences of (block) complexity kn+1 originally introduced by P. Arnoux and G. Rauzy. This vectorial division algorithm yields simultaneous rational approximations of the frequencies of the letters. We extend Berthe’s result to factors of Arnoux-Rauzy sequences by expressing both the frequencies and the number of factors with a given frequency, in terms of the ‘convergents’ obtained from the generalized continued fraction expansion of the frequencies of the letters.