Dynamical Systems
Dimension of sets of sequences defined in terms of recurrence of their prefixes
Comptes Rendus. Mathématique, Volume 343 (2006) no. 2, pp. 129-133.

Let Σ be the set of sequences $x=(xn)n⩾1$ of elements of $S={1,2,…,m}$ endowed with the usual ultrametric $d(x,y)=m−inf{k⩾0:xk+1≠yk+1}$. Let define

 $Rn(x)=inf{j>n:x1x2⋯xn=xjxj+1⋯xj+n−1}.$
We show that for any α and β such that $1⩽α⩽β⩽∞$ the Hausdorff dimension of the set is equal to 1.

Soit Σ l'ensemble des suites $x=(xn)n⩾1$ d'éléments de $S={1,2,…,m}$ muni de l'ultramétrique usuelle $d(x,y)=m−inf{k⩾0:xk+1≠yk+1}$. Posons

 $Rn(x)=inf{j>n:x1x2⋯xn=xjxj+1⋯xj+n−1}.$
Nous montrons que, quels que soient α et β tels que $1⩽α⩽β⩽∞$ l'ensemble a une dimension de Hausdorff égale à 1.

Accepted:
Published online:
DOI: 10.1016/j.crma.2006.05.005

Li Peng 1

1 Department of Mathematics, Wuhan University, Wuhan 430072, PR China
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.05.005/

[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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