Badly approximable vectors and fractals defined by conformal dynamical systems
Number Theory
2019-06-18 v2 Dynamical Systems
Abstract
We prove that if 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 . The same is true if is the radial Julia set of an irreducible meromorphic function (either rational or transcendental). The method of proof is to find subsets of 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.
Cite
@article{arxiv.1603.01467,
title = {Badly approximable vectors and fractals defined by conformal dynamical systems},
author = {Tushar Das and Lior Fishman and David Simmons and Mariusz Urbański},
journal= {arXiv preprint arXiv:1603.01467},
year = {2019}
}