Discrete Mathematics & Theoretical Computer Science,Vol 9, No 2 (2007)

摘要

A Parry number is a real number β >1 such thatthe Rényi β-expansion of 1 is finite or infiniteeventually periodic. If this expansion is finite, β is saidto 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 canonicalsubstitution associated with β-expansions, when β is a simpleParry number. In this paper we consider the case whereβ is a non-simple Parry number. We determine the structure of infinite leftspecial branches, which are an important tool for the computationof the complexity of uβ. These results allow in particular to obtain thefollowing characterization: the infinite word uβ is Sturmianif and only if β is a quadratic Pisot unit.