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

Soit Σ l'ensemble des suites x=(xn)n1 d'éléments de S={1,2,,m} muni de l'ultramétrique usuelle d(x,y)=minf{k0:xk+1yk+1}. Posons

Rn(x)=inf{j>n:x1x2xn=xjxj+1xj+n1}.
Nous montrons que, quels que soient α et β tels que 1αβ l'ensemble
Bα,β={xΣ:lim̲nlogRn(x)logn=α et lim¯nlogRn(x)logn=β}
a une dimension de Hausdorff égale à 1.

Let Σ be the set of sequences x=(xn)n1 of elements of S={1,2,,m} endowed with the usual ultrametric d(x,y)=minf{k0:xk+1yk+1}. Let define

Rn(x)=inf{j>n:x1x2xn=xjxj+1xj+n1}.
We show that for any α and β such that 1αβ the Hausdorff dimension of the set
Bα,β={xΣ:lim̲nlogRn(x)logn=α and lim¯nlogRn(x)logn=β}
is equal to 1.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.05.005

Li Peng 1

1 Department of Mathematics, Wuhan University, Wuhan 430072, PR China
@article{CRMATH_2006__343_2_129_0,
     author = {Li Peng},
     title = {Dimension of sets of sequences defined in terms of recurrence of their prefixes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {129--133},
     publisher = {Elsevier},
     volume = {343},
     number = {2},
     year = {2006},
     doi = {10.1016/j.crma.2006.05.005},
     language = {en},
}
TY  - JOUR
AU  - Li Peng
TI  - Dimension of sets of sequences defined in terms of recurrence of their prefixes
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 129
EP  - 133
VL  - 343
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2006.05.005
LA  - en
ID  - CRMATH_2006__343_2_129_0
ER  - 
%0 Journal Article
%A Li Peng
%T Dimension of sets of sequences defined in terms of recurrence of their prefixes
%J Comptes Rendus. Mathématique
%D 2006
%P 129-133
%V 343
%N 2
%I Elsevier
%R 10.1016/j.crma.2006.05.005
%G en
%F CRMATH_2006__343_2_129_0
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

Cité par Sources :

Commentaires - Politique