Badly approximable vectors and fractals defined by conformal dynamical systems

TL;DR

This paper demonstrates that badly approximable vectors have full Hausdorff dimension within certain fractal sets generated by conformal dynamical systems, extending to Julia sets of meromorphic functions, using measures with specific regularity properties.

Contribution

It establishes the full Hausdorff dimension of badly approximable vectors in fractals from conformal systems and Julia sets, and shows hyperplane diffuse sets support absolutely decaying measures.

Findings

Badly approximable vectors have full Hausdorff dimension in limit sets of conformal iterated function systems.

The same holds for radial Julia sets of irreducible meromorphic functions.

Every hyperplane diffuse set supports an absolutely decaying measure.

Abstract

We prove that if $J$ is the limit set of an irreducible conformal iterated function system (with either finite or countably infinite alphabet), then the badly approximable vectors form a set of full Hausdorff dimension in $J$. The same is true if $J$ is the radial Julia set of an irreducible meromorphic function (either rational or transcendental). The method of proof is to find subsets of $J$ that support absolutely friendly and Ahlfors regular measures of large dimension. In the appendix to this paper, we answer a question of Broderick, Kleinbock, Reich, Weiss, and the second-named author ('12) by showing that every hyperplane diffuse set supports an absolutely decaying measure.