Infinite special branches in words associated with beta-expansions

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.