Inhomogeneous potentials, Hausdorff dimension and shrinking targets

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.Comment: 36 pages. Corrected and reorganised following referee's report. Accepted for publication in Annales Henri Lebesgu

Hausdorff dimension of random limsup sets

We prove bounds for the almost sure value of the Hausdorff dimension of the limsup set of a sequence of balls in $\mathbf{R}^d$ whose centres are independent, identically distributed random variables. The formulas obtained involve the rate of decrease of the radii of the balls and multifractal properties of the measure according to which the balls are distributed, and generalise formulas that are known to hold for particular classes of measures.Comment: 26 pages, 2 figures; v2: Minor correction

Bernoulli Convolutions and 1D Dynamics

We describe a family $\phi_{\lambda}$ of dynamical systems on the unit interval which preserve Bernoulli convolutions. We show that if there are parameter ranges for which these systems are piecewise convex, then the corresponding Bernoulli convolution will be absolutely continuous with bounded density. We study the systems $\phi_{\lambda}$ and give some numerical evidence to suggest values of $\lambda$ for which $\phi_{\lambda}$ may be piecewise convex.Comment: 18 page