Return words in Sturmian and episturmian words

Abstract

Considering each occurrence of a word w in a recurrent infinite word, we define the set of return words of w to be the set of all distinct words beginning with an occurrence of w and ending exactly just before the next occurrence of w in the infinite word. We give a simpler proof of the recent result (of the second author) that an infinite word is Sturmian if and only if each of its factors has exactly two return words in it. Then, considering episturmian infinite words, which are a natural generalization of Sturmian words, we study the position of the occurrences of any factor in such infinite words and we determinate the return words. At last, we apply these results in order to get a kind of balance property of episturmian words and to calculate the recurrence function of these words.