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# Mathematical Research Letters

## Volume 25 (2018)

### Number 2

### Badly approximable vectors and fractals defined by conformal dynamical systems

Pages: 437 – 467

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

#### Authors

#### 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.

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

Accepted 16 January 2017

Published 5 July 2018