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 uβ 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 uβ. These results allow in particular to obtain the
following characterization: the infinite word uβ is Sturmian
if and only if β is a quadratic Pisot unit.
Full Text: PostScript PDF