Inhomogeneous potentials, Hausdorff dimension and shrinking targets

TL;DR

This paper develops a framework linking inhomogeneous potentials to Hausdorff dimension, enabling the estimation of dimensions of complex sets like limsup-sets and shrinking target sets in dynamical systems.

Contribution

It introduces a new class of $G_\delta$-sets with large Hausdorff dimension, related to inhomogeneous potentials, and applies this to estimate dimensions of dynamically defined sets.

Findings

Calculated Hausdorff dimension of certain limsup-sets.

Estimated dimensions of shrinking target sets in quadratic dynamical systems.

Established a connection between inhomogeneous energies and set dimensions.

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.