Infinite special branches in words associated with beta-expansions

• Main Authors: Christiane Frougny, Zuzana Masáková, Edita Pelantová
• Format: Article
• Language: English
• Published: Discrete Mathematics & Theoretical Computer Science 2007-05-01
• Series: Discrete Mathematics & Theoretical Computer Science
• Online Access: http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/658

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.