Sequences with constant number of return words

Lubomira Balkova; Edita Pelantova; Wolfgang Steiner

paper

Sequences with constant number of return words

arXiv:math/0608603

Abstract

An infinite word has the property $R_{m}$ if every factor has exactly $m$ return words. Vuillon showed that $R_{2}$ characterizes Sturmian words. We prove that a word satisfies $R_{m}$ if its complexity function is $(m - 1) n + 1$ and if it contains no weak bispecial factor. These conditions are necessary for $m = 3$ , whereas for $m = 4$ the complexity function need not be $3 n + 1$ . New examples of words satisfying $R_{m}$ are given by words related to digital expansions in real bases.