*-sturmian words and complexity

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

Journal de théorie des nombres de Bordeaux (2003)

  • Volume: 15, Issue: 3, page 767-804
  • ISSN: 1246-7405

Abstract

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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 ) ≤ 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 values, namely, p ( n ) ≥ 2 n 1 - ϵ holds for some * -Sturmian words; however the topological entropy of any * -Sturmian word is zero.

How to cite

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Nakashima, Izumi, Tamura, Jun-Ichi, and Yasutomi, Shin-Ichi. "*-sturmian words and complexity." Journal de théorie des nombres de Bordeaux 15.3 (2003): 767-804. <http://eudml.org/doc/249073>.

@article{Nakashima2003,
abstract = {We give analogs of the complexity $p(n)$ and of Sturmian words which are called respectively the $\ast $-complexity $p_\ast (n)$ and $\ast $-Sturmian words. We show that the class of $\ast $-Sturmian words coincides with the class of words satisfying $p_\ast (n) \le n + 1$, and we determine the structure of $\ast $-Sturmian words. For a class of words satisfying $p_\ast (n) = n + 1$, 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 values, namely, $p(n) \ge 2^\{n^\{1- \epsilon \}\}$ holds for some $\ast $-Sturmian words; however the topological entropy of any $\ast $-Sturmian word is zero.},
author = {Nakashima, Izumi, Tamura, Jun-Ichi, Yasutomi, Shin-Ichi},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Sturmian words; complexity; approximation of rational lines; infinite occurrences},
language = {eng},
number = {3},
pages = {767-804},
publisher = {Université Bordeaux I},
title = {*-sturmian words and complexity},
url = {http://eudml.org/doc/249073},
volume = {15},
year = {2003},
}

TY - JOUR
AU - Nakashima, Izumi
AU - Tamura, Jun-Ichi
AU - Yasutomi, Shin-Ichi
TI - *-sturmian words and complexity
JO - Journal de théorie des nombres de Bordeaux
PY - 2003
PB - Université Bordeaux I
VL - 15
IS - 3
SP - 767
EP - 804
AB - We give analogs of the complexity $p(n)$ and of Sturmian words which are called respectively the $\ast $-complexity $p_\ast (n)$ and $\ast $-Sturmian words. We show that the class of $\ast $-Sturmian words coincides with the class of words satisfying $p_\ast (n) \le n + 1$, and we determine the structure of $\ast $-Sturmian words. For a class of words satisfying $p_\ast (n) = n + 1$, 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 values, namely, $p(n) \ge 2^{n^{1- \epsilon }}$ holds for some $\ast $-Sturmian words; however the topological entropy of any $\ast $-Sturmian word is zero.
LA - eng
KW - Sturmian words; complexity; approximation of rational lines; infinite occurrences
UR - http://eudml.org/doc/249073
ER -

References

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