Inhomogeneous potentials, Hausdorff dimension and shrinking targets

Abstract

Generalising a construction of Falconer, we consider classes of G -subsets of R d with the property that sets belonging to the class have large Hausdor 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 Hausdor 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 ! 1, dimHf y : jT n a (x) yj < n innitely often g = 1 ;for almost every x P [1 a; 1], where Ta is a quadratic map with a in a set of parameters described by Benedicks and Carleson.

Uniform Random Covering Problems

“…Suppose η is a Borel measure, and let 0 Proof. The proof follows the proof of Lemma 1.1 (or Lemma 5.1) in Persson [14].…”

Hausdorff dimension of limsup sets of isotropic rectangles in Heisenberg groups

“…Since [15], there have been many developments in the study of dimensional properties of random limsup sets in metric spaces. For various results, see [9,13,14,17,20,21,24,25,26]. Together, the dimension results in these papers yield the almost-sure values of Hausdorff dimensions of random limsup sets in varying scenarios depending on properties of the ambient space, the sets generating the limsup sets and the measure according to which the sets are distributed.…”

Hausdorff dimension of random limsup sets

“…The Hausdorff dimension of sets of this kind was studied for particular types of dynamical systems, see for instance . For the dynamical systems studied in those papers, the problem of determining the Hausdorff dimension of the set is rather close to that of this paper.…”