Inhomogeneous potentials, Hausdorff dimension and shrinking targets

Abstract

Generalising a construction of Falconer, we consider classes of $G_\delta$-subsets of $\mathbb{R}^d$ 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 $\alpha \geq 1$, \[ \mathrm{dim}_\mathrm{H}\, \{ \, y : | T_a^n (x) - y| < n^{-\alpha} \text{ infinitely often} \, \} = \frac{1}{\alpha}, \] for almost every $x \in [1-a,1]$, where $T_a$ is a quadratic map with $a$ in a set of parameters described by Benedicks and Carleson.