Sequences with constant number of return words

Lubomira Balkova (1) , Edita Pelantová (1) , Wolfgang Steiner (2)

1 Doppler Institute
2 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

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 $3n+1$. New examples of words satisfying $R_m$ are given by words related to digital expansions in real bases.

Mots clés: return word; complexity; beta-integers

Domaines: Mathématique discrète [cs.DM]; Combinatoire [math.CO]

Lubomira Balkova, Edita Pelantová, Wolfgang Steiner. Sequences with constant number of return words. Monatshefte für Mathematik, 2008, 155 (3-4), pp.251-263. ⟨10.1007/s00605-008-0001-2⟩. ⟨hal-00089863v4⟩