Proceedings of the Workshop on Fibonacci Words Turku , September 2006

Recently, the notion of return words has appeared in various branches of mathematics, for instance in Symbolic Dynamical Systems and Number Theory. This mathematical tool has been introduced by Durand [2] in order to obtain a nice characterization of primitive substitutive sequences. Roughly speaking, for a given factor w of the infinite word u, a return word of w is a segment between two successive occurrences of the factor w. Using return words, Vuillon in [9] has found a new equivalent definition of sturmian words. He has shown that an infinite word u over a biliteral alphabet is sturmian if and only if any factor of u has two return words. A natural generalization of sturmian words to a multiliteral alphabet is provided by Arnoux-Rauzy words. Justin and Vuillon have proved that ArnouxRauzy words of order m have for every factor exactlym return words. Here,we present three very simple ideas which enable to determine easily the number of return words in some infinite words. • The first observation is that for study of return words of an infinite uniformly recurrent word u, it suffices to limit the considerations to bispecial factors of u. • We show the importance of Rauzy graphs for determination of the number of return words. • In case of infinite words invariant under a substitution, we will make use of the relation between return words of a factor w and return words of its image.