Repetitions in beta-integers

Classical crystals are solid materials containing arbitrarily long periodic repetitions of a single motif. In this paper, we study the maximal possible repetition of the same motif occurring in beta-integers -- one dimensional models of quasicrystals. We are interested in beta-integers realizing only a finite number of distinct distances between neighboring elements. In such a case, the problem may be reformulated in terms of combinatorics on words as a study of the index of infinite words coding beta-integers. We will solve a particular case for beta being a quadratic non-simple Parry number.Comment: 11 page

Palindromic complexity of infinite words associated with non-simple Parry numbers

We study the palindromic complexity of infinite words uβ, the fixed points of the substitution over a binary alphabet, φ(0) = 0a1, φ(1) = 0b1, with a - 1 ≥ b ≥ 1, which are canonically associated with quadratic non-simple Parry numbers β

Theoretical Informatics and Applications Will be set by the publisher Informatique Théorique et Applications PALINDROMIC COMPLEXITY OF INFINITE WORDS ASSOCIATED WITH NON-SIMPLE PARRY NUMBERS

Abstract. We study the palindromic complexity of infinite words u β , the fixed points of the substitution over a binary alphabet, ϕ(0) = 0 a 1, ϕ(1) = 0 b 1, with a − 1 ≥ b ≥ 1, which are canonically associated with quadratic non-simple Parry numbers β