Geodesic
Infinite special branches in words associated with beta-expansions
Discrete mathematics & theoretical computer science, Tome 9 (2007) no. 2.

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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. 
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     title = {Infinite special branches in words associated with beta-expansions},
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Frougny, Christiane; Masáková, Zuzana; Pelantová, Edita. Infinite special branches in words associated with beta-expansions. Discrete mathematics & theoretical computer science, Tome 9 (2007) no. 2. doi : 10.46298/dmtcs.415. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.415/

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