Mathematical Research Letters

Volume 25 (2018)

Number 2

Badly approximable vectors and fractals defined by conformal dynamical systems

Pages: 437 – 467

DOI: http://dx.doi.org/10.4310/MRL.2018.v25.n2.a5

Authors

Tushar Das (Department of Mathematics & Statistics, University of Wisconsin, La Crosse, Wi., U.S.A.)

Lior Fishman (Department of Mathematics, University of North Texas, Denton, Tx., U.S.A.)

David Simmons (Department of Mathematics, University of York, Heslington, York, United Kingdom)

Mariusz Urbański (Department of Mathematics, University of North Texas, Denton, Tx., U.S.A.)

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.

Keywords

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.

Full Text (PDF format)

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