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

Li Peng

doi:10.1016/j.crma.2006.05.005

Dynamical Systems

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

Jean-Pierre Kahane

Li Peng ¹

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

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

Abstract (Original language)
Résumé

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.

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.

Article information
Cite this article

Received: 2006-02-21
Accepted: 2006-03-14
Published online: 2006-06-27

DOI: 10.1016/j.crma.2006.05.005

Author's affiliations:

Li Peng ¹

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

Li Peng. Dimension of sets of sequences defined in terms of recurrence of their prefixes. Comptes Rendus. Mathématique, Volume 343 (2006) no. 2, pp. 129-133. doi: 10.1016/j.crma.2006.05.005

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     title = {Dimension of sets of sequences defined in terms of recurrence of their prefixes},
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     pages = {129--133},
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     publisher = {Elsevier},
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     number = {2},
     doi = {10.1016/j.crma.2006.05.005},
     language = {en},
}

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References
Cited by

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

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

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