Badly approximable vectors and fractals defined by conformal dynamical systems

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.