Return words in fixed points of substitutions

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

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

We present three simple ideas which enable to determine easily the number of return words in some infinite words. Using these ideas, we give a new and very short proof of the fact that each factor of an Arnoux-Rauzy word of order $m$ has exactly $m$ return words. We describe the structure of return words for the Thue-Morse sequence and focus on infinite words which are associated to simple Parry numbers.

Mots clés: return word; factor; Thue-Morse sequence; beta-integers

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

Lubomira Balkova, Edita Pelantová, Wolfgang Steiner. Return words in fixed points of substitutions. 2007. ⟨hal-00089863v3⟩