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Comptes Rendus Mathématique
Volume 355, n° 7
pages 738-743 (juillet 2017)
Doi : 10.1016/j.crma.2017.05.014
Received : 27 November 2016 ;  accepted : 31 May 2017
Dynamical covering problems on the triadic Cantor set
Problèmes de recouvrement dynamique sur les ensembles de Cantor triadiques
 

Bao-Wei Wang , Jun Wu , Jian Xu
 School of Mathematics and Statistics, Huazhong University of Science and Technology, 430074 Wuhan, China 

Abstract

In this note, we consider the metric theory of the dynamical covering problems on the triadic Cantor set  . More precisely, let   be the natural map on  , μ the standard Cantor measure and   a given point. We consider the size of the set of points in   which can be well approximated by the orbit   of  , namely the set
D(x0,φ):={y∈K:|Tnx0−y|<φ(n)for infinitely manyn∈N}, where φ is a positive function defined on  . It is shown that for μ almost all  , the Hausdorff measure of   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  .

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Résumé

Nous considérons dans cette Note la théorie métrique des recouvrements dynamiques dans l'ensemble de Cantor triadique  . Plus précisément, soit   l'application naturelle sur  , μ la mesure de Cantor standard et   un point donné. Nous considérons la mesure de l'ensemble des points de   qui peuvent être bien approchés par l'orbite   de  , 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  . Nous montrons que pour μ -presque tout   la mesure de Hausdorff de   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.

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