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

$$R_n(x)=\inf \{j>n:x_1{x}_2\cdots x_n=x_j{x}_{j+1}\cdots x_{j+n-1}\}.$$

Nous montrons que, quels que soient α et β tels que $1⩽\alpha ⩽\beta ⩽\infty$ l'ensemble

$$B_{\alpha ,\beta }=\{x\in \Sigma :\underset{n\to \infty }{\underline{\lim }}\frac{\log R_n(x)}{\log n}=\alpha \text{ et }\underset{n\to \infty }{\overline{\lim }}\frac{\log R_n(x)}{\log n}=\beta \}$$

a une dimension de Hausdorff égale à 1.

Reçu le : 2006-02-21
Accepté le : 2006-03-14
Publié le : 2006-06-27
DOI : 10.1016/j.crma.2006.05.005

@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},
}

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VL  - 343
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crma.2006.05.005
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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/