Mathematical Research Letters
Volume 25 (2018)
Badly approximable vectors and fractals defined by conformal dynamical systems
Pages: 437 – 467
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.
diophantine approximation, badly approximable vectors, conformal dynamical systems, Hausdorff dimension, iterated function system, meromorphic function, radial Julia set, hyperbolic dimension, elliptic function
2010 Mathematics Subject Classification
Primary 11J83, 37F10. Secondary 37C45, 37F35.
The first-named author was supported in part by a 2016-2017 Faculty Research Grant from the University of Wisconsin–La Crosse. The secondnamed author was supported in part by the Simons Foundation grant #245708. The third-named author was supported in part by the EPSRC Programme Grant EP/J018260/1. The fourth-named author was supported in part by the NSF grant DMS-1361677. The authors thank Barak Weiss and the anonymous referee for helpful comments.
Received 17 March 2016
Published 5 July 2018