no. 7
Number theory/Dynamical systems
Dynamical covering problems on the triadic Cantor set
[Problèmes de recouvrement dynamique sur les ensembles de Cantor triadiques]

Présenté par : the Editorial Board

Wang, Bao-Wei1 ; Wu, Jun1 ; Xu, Jian1
1 School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China
Comptes Rendus. Mathématique, Tome 355 (2017) no. 7, pp. 738-743.

Nous considérons dans cette Note la théorie métrique des recouvrements dynamiques dans l'ensemble de Cantor triadique K. Plus précisément, soit Tx=3x(mod1) l'application naturelle sur K, μ la mesure de Cantor standard et x0∈K un point donné. Nous considérons la mesure de l'ensemble des points de K qui peuvent être bien approchés par l'orbite {Tnx0}n≥1 de x0, c'est-à-dire l'ensemble

D(x0,φ):={y∈K:|Tnx0−y|<φ(n)pour une infinité den∈N},
où φ est une fonction positive définie sur N. Nous montrons que pour μ-presque tout x0∈K la mesure de Hausdorff de D(x0,φ) est soit zéro, soit pleine, selon la convergence ou la divergence d'une certaine série. Notre démonstration fournit en passant une contre-partie inhomogène au travail de Levesley, Salp et Velani sur une question de Mahler relative à l'approximation rationnelle des points de l'ensemble de Cantor.

In this note, we consider the metric theory of the dynamical covering problems on the triadic Cantor set K. More precisely, let Tx=3x(mod1) be the natural map on K, μ the standard Cantor measure and x0∈K a given point. We consider the size of the set of points in K which can be well approximated by the orbit {Tnx0}n≥1 of x0, namely the set

D(x0,φ):={y∈K:|Tnx0−y|<φ(n)for infinitely manyn∈N},
where φ is a positive function defined on N. It is shown that for μ almost all x0∈K, the Hausdorff measure of D(x0,φ) is either zero or full depending upon the convergence or divergence of a certain series. Among the proof, as a byproduct, we obtain an inhomogeneous counterpart of Levesley, Salp and Velani's work on a Mahler's question about the Diophantine approximation on the Cantor set K.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2017.05.014
Wang, Bao-Wei 1 ; Wu, Jun 1 ; Xu, Jian 1

1 School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China 
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Wang, Bao-Wei; Wu, Jun; Xu, Jian. Dynamical covering problems on the triadic Cantor set. Comptes Rendus. Mathématique, Tome 355 (2017) no. 7, pp. 738-743. doi : 10.1016/j.crma.2017.05.014. http://www.numdam.org/articles/10.1016/j.crma.2017.05.014/

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