Infinite Special Branches in Words Associated with Beta-Expansions

A Parry number is a real number β > 1 such that the Renyi 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.

[1]  A. Rényi.  Representations for real numbers and their ergodic properties , 1957 .

[2]  Christiane Frougny,et al.  Palindromic complexity of infinite words associated with simple Parry numbers , 2006 .

[3]  Christiane Frougny.  Confluent Linear Numeration Systems , 1992, Theor. Comput. Sci..

[4]  C. Mauduit,et al.  Substitutions in dynamics, arithmetics, and combinatorics , 2002 .

[5]  Christiane Frougny,et al.  Complexity of infinite words associated with beta-expansions , 2004, RAIRO Theor. Informatics Appl..

[6]  W. Parry.  On theβ-expansions of real numbers , 1960 .

[7]  三浦 省五,et al.  Oral Communication , 2007 .

[8]  Randolph B. Tarrier,et al.  Groups , 1973 .

[9]  Boris Solomyak,et al.  Finite beta-expansions , 1992, Ergodic Theory and Dynamical Systems.

[10]  Julien Cassaigne,et al.  Complexité et facteurs spéciaux , 1997 .

[11]  Christiane Frougny,et al.  Additive and multiplicative properties of point sets based on beta-integers , 2003, Theor. Comput. Sci..

[12]  Giuseppe Pirillo,et al.  Episturmian words and some constructions of de Luca and Rauzy , 2001, Theor. Comput. Sci..

[13]  Julien Cassaigne,et al.  Special Factors of Sequences with Linear Subword Complexity , 1995, Developments in Language Theory.

[14]  Jean-Paul Allouche,et al.  Sur la complexite des suites in nies , 1994 .

[15]  M. Lothaire.  Algebraic Combinatorics on Words , 2002 .

[16]  Robert Tijdeman.  On the minimal complexity of infinite words , 1999 .

[17]  Tero Harju,et al.  Combinatorics on Words , 2004 .

[18]  K. Schmidt,et al.  On Periodic Expansions of Pisot Numbers and Salem Numbers , 1980 .