On low-complexity bi-infinite words and their factors

Alex Heinis

Displaying similar documents to “On low-complexity bi-infinite words and their factors”

*-sturmian words and complexity

Izumi Nakashima, Jun-Ichi Tamura, Shin-Ichi Yasutomi (2003)

Journal de théorie des nombres de Bordeaux

Similarity:

We give analogs of the complexity $p (n)$ and of Sturmian words which are called respectively the $*$ -complexity $p_{*} (n)$ and $*$ -Sturmian words. We show that the class of $*$ -Sturmian words coincides with the class of words satisfying $p_{*} (n) \leq n + 1$ , and we determine the structure of $*$ -Sturmian words. For a class of words satisfying $p_{*} (n) = n + 1$ , we give a general formula and an upper bound for $p (n)$ . Using this general formula, we give explicit formulae for $p (n)$ for some words belonging to this class. In general, $p (n)$ can take large...

Hereditary properties of words

József Balogh, Béla Bollobás (2005)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Similarity:

Let $𝒫$ be a hereditary property of words, i.e., an infinite class of finite words such that every subword (block) of a word belonging to $𝒫$ is also in $𝒫$ . Extending the classical Morse-Hedlund theorem, we show that either $𝒫$ contains at least $n + 1$ words of length $n$ for every $n$ or, for some $N$ , it contains at most $N$ words of length $n$ for every $n$ . More importantly, we prove the following quantitative extension of this result: if $𝒫$ has $m \leq n$ words of length $n$ then, for every $k \geq n + m$ , it contains at most...

Complexity of infinite words associated with beta-expansions

Christiane Frougny, Zuzana Masáková, Edita Pelantová (2004)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Similarity:

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...

Substitution invariant sturmian bisequences

Bruno Parvaix (1999)

Journal de théorie des nombres de Bordeaux

Similarity:

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.

Episturmian words: a survey

Amy Glen, Jacques Justin (2009)

RAIRO - Theoretical Informatics and Applications

Similarity:

In this paper, we survey the rich theory of infinite which generalize to any finite alphabet, in a rather resembling way, the well-known family of on two letters. After recalling definitions and basic properties, we consider that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their...

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

Similarity:

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...

On Christoffel classes

Jean-Pierre Borel, Christophe Reutenauer (2006)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

Similarity:

We characterize conjugation classes of Christoffel words (equivalently of standard words) by the number of factors. We give several geometric proofs of classical results on these words and sturmian words.