Infinite special branches in words associated with beta-expansions

A Parry number is a real number beta > 1 such that the Renyi beta-expansion of 1 is finite or in finite eventually periodic. If this expansion is finite, beta 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(beta) of the canonical substitution associated with beta-expansions, when beta is a simple Parry number. In this paper we consider the case where beta 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(beta). These results allow in particular to obtain the following characterization: the infinite word u(beta) is Sturmian if and only if beta is a quadratic Pisot unit.