*-sturmian words and complexity

Izumi Nakashima; Jun-Ichi Tamura; Shin-Ichi Yasutomi

Displaying similar documents to “*-sturmian words and complexity”

On low-complexity bi-infinite words and their factors

Alex Heinis (2001)

Journal de théorie des nombres de Bordeaux

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In this paper we study bi-infinite words on two letters. We say that such a word has stiffness $k$ if the number of different subwords of length $n$ equals $n + k$ for all $n$ sufficiently large. The word is called $k$ -balanced if the numbers of occurrences of the symbol a in any two subwords of the same length differ by at most $k$ . In the present paper we give a complete description of the class of bi-infinite words of stiffness $k$ and show that the number of subwords of length $n$ from this class has...

Substitution invariant sturmian bisequences

Bruno Parvaix (1999)

Journal de théorie des nombres de Bordeaux

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We prove that a Sturmian bisequence, with slope $α$ and intercept $ρ$ , is fixed by some non-trivial substitution if and only if $α$ is a Sturm number and $ρ$ belongs to $ℚ (α)$ . We also detail a complementary system of integers connected with Beatty bisequences.

Complexity of infinite words associated with beta-expansions

Christiane Frougny, Zuzana Masáková, Edita Pelantová (2004)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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We study the complexity of the infinite word $u_{β}$ associated with the Rényi expansion of $1$ in an irrational base $β > 1$ . When $β$ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $ℂ (n) = n + 1$ . For $β$ such that $d_{β} (1) = t_{1} t_{2} \dots t_{m}$ is finite we provide a simple description of the structure of special factors of the word $u_{β}$ . When $t_{m} = 1$ we show that $ℂ (n) = (m - 1) n + 1$ . In the cases when $t_{1} = t_{2} = \dots = t_{m - 1}$ or $t_{1} > max {t_{2}, \dots, t_{m - 1}}$ we show that the first difference of the complexity function $ℂ (n + 1) - ℂ (n)$ takes value in ${m - 1, m}$ for every $n$ , and consequently...

Imbalances in Arnoux-Rauzy sequences

Julien Cassaigne, Sébastien Ferenczi, Luca Q. Zamboni (2000)

Annales de l'institut Fourier

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In a 1982 paper Rauzy showed that the subshift $(X, T)$ generated by the morphism $1 \mapsto 12$ , $2 \mapsto 13$ and $3 \mapsto 1$ is a natural coding of a rotation on the two-dimensional torus $𝕋^{2}$ , i.e., is measure-theoretically conjugate to an exchange of three fractal domains on a compact set in $ℝ^{2},$ each domain being translated by the same vector modulo a lattice. It was believed more generally that each sequence of block complexity $2 n + 1$ satisfying a combinatorial criterion known as the $☆$ condition of Arnoux and Rauzy codes the orbit...

Combinatorial properties of infinite words associated with cut-and-project sequences

Louis-Sébastien Guimond, Zuzana Masáková, Edita Pelantová (2003)

Journal de théorie des nombres de Bordeaux

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The aim of this article is to study certain combinatorial properties of infinite binary and ternary words associated to cut-and-project sequences. We consider here the cut-and-project scheme in two dimensions with general orientation of the projecting subspaces. We prove that a cut-and-project sequence arising in such a setting has always either two or three types of distances between adjacent points. A cut-and-project sequence thus determines in a natural way a symbolic sequence (infinite...