Badly approximable vectors and fractals defined by conformal dynamical systems

Tushar Das; Lior Fishman; David Simmons; Mariusz Urbański

doi:10.4310/MRL.2018.v25.n2.a5

Badly approximable vectors and fractals defined by conformal dynamical systems

Tushar Das, Lior Fishman, David Simmons, Mariusz Urbański

Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Journal	Mathematical Research Letters
Early online date	5 Jul 2018
DOIs	https://doi.org/10.4310/MRL.2018.v25.n2.a5
Publication status	E-pub ahead of print - 5 Jul 2018

Bibliographical note

This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details

Keywords

math.NT
math.DS

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10.4310/MRL.2018.v25.n2.a5

Badly approximable vectors
This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy. Further copying may not be permitted; contact the publisher for details
Accepted author manuscript, 326 KB

https://arxiv.org/abs/1603.01467

Programme Grant-New Frameworks in metric Number Theory
Velani, S. & Beresnevich, V.
EPSRC
1/06/12 → 30/11/18
Project: Research project (funded) › Research

Cite this

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AU - Fishman, Lior

AU - Simmons, David

AU - Urbański, Mariusz

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N2 - 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.

AB - 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.

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M3 - Article

SN - 1073-2780

JO - Mathematical Research Letters

JF - Mathematical Research Letters

ER -

Badly approximable vectors and fractals defined by conformal dynamical systems

Abstract

Bibliographical note

Keywords

Access to Document

Projects

Programme Grant-New Frameworks in metric Number Theory

Cite this