Inhomogeneous potentials, Hausdorff dimension and shrinking targets
[Potentiels inhomogènes, dimension de Hausdorff et cibles rétrécissantes]
Annales Henri Lebesgue, Tome 2 (2019), pp. 1-37.

Généralisant une construction de Falconer, nous considérons des classes de sous-ensembles G δ de d qui sont fermées sous intersections dénombrables et dont les ensembles ont une grande dimension de Hausdorff. Nous relions ces classes à certains potentiels et énergies inhomogènes, fournissant ainsi des outils permettant de déterminer si un ensemble appartient à l’une des classes.

Comme applications de cette théorie, nous calculons, ou du moins estimons, la dimension de Hausdorff d’ensembles de limsup générés aléatoirement et d’ensembles apparaissant dans le cadre des cibles rétrécissantes en systèmes dynamiques. Par exemple, nous prouvons que pour α1,

dim H {y:|T a n (x)-y|<n -α infinimentsouvent}=1 α,

pour presque tout x[1-a,1], où T a est une application quadratique pour a dans l’ensemble de paramètres de Benedicks et Carleson.

Generalising a construction of Falconer, we consider classes of G δ -subsets of d with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We relate these classes to some inhomogeneous potentials and energies, thereby providing some useful tools to determine if a set belongs to one of the classes.

As applications of this theory, we calculate, or at least estimate, the Hausdorff dimension of randomly generated limsup-sets, and sets that appear in the setting of shrinking targets in dynamical systems. For instance, we prove that for α1,

dim H {y:|T a n (x)-y|<n -α infinitelyoften}=1 α,

for almost every x[1-a,1], where T a is a quadratic map with a in a set of parameters described by Benedicks and Carleson.

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DOI : 10.5802/ahl.15
Classification : 37C45, 37E05, 28A78, 28A05, 60D05
Mots clés : Hausdorff dimension, limsup sets, potentials
Persson, Tomas 1

1 Centre for Mathematical Sciences Lund University, Box 118 221 00 Lund (Sweden)
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Persson, Tomas. Inhomogeneous potentials, Hausdorff dimension and shrinking targets. Annales Henri Lebesgue, Tome 2 (2019), pp. 1-37. doi : 10.5802/ahl.15. http://archive.numdam.org/articles/10.5802/ahl.15/

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