Badly approximable vectors and fractals defined by conformal dynamical systems

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JournalMathematical Research Letters
DateAccepted/In press - 16 Jan 2017
DateE-pub ahead of print (current) - 5 Jul 2018
Early online date5/07/18
Original languageEnglish

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.

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  • math.NT, math.DS

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