Work

Badly approximable vectors and fractals defined by conformal dynamical systems

Year: 2018

Type: article

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.

Source: Mathematical Research Letters

DOI: 10.4310/mrl.2018.v25.n2.a5

Authors: Tushar Das, Lior Fishman, David Simmons, Mariusz Urbański

Institutions: University of Wisconsin–La Crosse, University of North Texas, University of York

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, Simons Foundation, Engineering and Physical Sciences Research Council

Award: G3132472811