Dimension of sets of sequences defined in terms of recurrence of their prefixes

Peng, Li

doi:10.1016/j.crma.2006.05.005

Dynamical Systems

Dimension of sets of sequences defined in terms of recurrence of their prefixes
[Dimension d'ensembles de suites dont les préfixes réapparaissent en un temps prescrit]

Présenté par : Jean-Pierre Kahane

Peng, Li ¹

¹ Department of Mathematics, Wuhan University, Wuhan 430072, PR China

Comptes Rendus. Mathématique, Tome 343 (2006) no. 2, pp. 129-133.

Résumé
Abstract

Soit Σ l'ensemble des suites $x = {(x_{n})}_{n ⩾ 1}$ d'éléments de $S = {1, 2, \dots, m}$ muni de l'ultramétrique usuelle $d (x, y) = m^{- \inf {k ⩾ 0 : x_{k + 1} \neq y_{k + 1}}}$ . Posons

R_{n} (x) = \inf {j > n : x_{1} x_{2} \dots x_{n} = x_{j} x_{j + 1} \dots x_{j + n - 1}} .

Nous montrons que, quels que soient α et β tels que

1 ⩽ α ⩽ β ⩽ \infty

l'ensemble

B_{α, β} = {x \in Σ : \underset{n \to \infty}{\lim_{̲}} \frac{\log R_{n} (x)}{\log n} = α et \underset{n \to \infty}{\lim^{¯}} \frac{\log R_{n} (x)}{\log n} = β}

a une dimension de Hausdorff égale à 1.

Let Σ be the set of sequences $x = {(x_{n})}_{n ⩾ 1}$ of elements of $S = {1, 2, \dots, m}$ endowed with the usual ultrametric $d (x, y) = m^{- \inf {k ⩾ 0 : x_{k + 1} \neq y_{k + 1}}}$ . Let define

R_{n} (x) = \inf {j > n : x_{1} x_{2} \dots x_{n} = x_{j} x_{j + 1} \dots x_{j + n - 1}} .

We show that for any α and β such that

1 ⩽ α ⩽ β ⩽ \infty

the Hausdorff dimension of the set

B_{α, β} = {x \in Σ : \underset{n \to \infty}{\lim_{̲}} \frac{\log R_{n} (x)}{\log n} = α and \underset{n \to \infty}{\lim^{¯}} \frac{\log R_{n} (x)}{\log n} = β}

is equal to 1.

Reçu le : 2006-02-21
Accepté le : 2006-03-14
Publié le : 2006-06-27

DOI : 10.1016/j.crma.2006.05.005

Affiliations des auteurs :

Peng, Li ¹

¹ Department of Mathematics, Wuhan University, Wuhan 430072, PR China

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     title = {Dimension of sets of sequences defined in terms of recurrence of their prefixes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {129--133},
     publisher = {Elsevier},
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     number = {2},
     year = {2006},
     doi = {10.1016/j.crma.2006.05.005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2006.05.005/}
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Peng, Li. Dimension of sets of sequences defined in terms of recurrence of their prefixes. Comptes Rendus. Mathématique, Tome 343 (2006) no. 2, pp. 129-133. doi : 10.1016/j.crma.2006.05.005. http://www.numdam.org/articles/10.1016/j.crma.2006.05.005/

Bibliographie
Cité par

[1] Feng, D.J.; Wu, J. The Hausdorff dimension of recurrent sets in symbolic spaces, Nonlinearity, Volume 14 (2001) no. 1, pp. 81-85

[2] Ornstein, D.S.; Weiss, B. Entropy and data compression schemes, IEEE Trans. Inform. Theory, Volume 39 (1993) no. 1, pp. 78-83

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