### Infinite special branches in words associated with beta-expansions

*Christiane Frougny, Zuzana Masáková, Edita Pelantová*

#### Abstract

A Parry number is a real number β >1 such that
the Rényi β-expansion of 1 is finite or infinite
eventually periodic. If this expansion is finite, β is said
to be a

*simple*Parry number. Remind that any Pisot number is a Parry number. In a previous work we have determined the complexity of the fixed point u_{β}of the canonical substitution associated with β-expansions, when β is a simple Parry number. In this paper we consider the case where β is a non-simple Parry number. We determine the structure of infinite left special branches, which are an important tool for the computation of the complexity of u_{β}. These results allow in particular to obtain the following characterization: the infinite word u_{β}is Sturmian if and only if β is a quadratic Pisot unit.Full Text: PostScript PDF