2016/03/31 by Tushar Das, Lior Fishman, David Simmons +1 · 6 citations
Mathematics · #Analytic and geometric function theory #Cantor set #Combinatorics #Conformal map #Countable set #Fractal #Function (biology) #Hausdorff dimension #Hyperplane #Iterated function system #Julia set #Limit (mathematics) #Limit set #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Measure (data warehouse) #Meromorphic function #Pure mathematics #Set (abstract data type) #Stochastic processes and statistical mechanics #Transcendental number #math.DS #math.NT
paper · pdf · doi:10.4310/mrl.2018.v25.n2.a5
published in Mathematical Research Letters 25(2), 437-467 (International Press of Boston)
arxiv created 2016/10/30 · openalex publication_date 2018/01/01 · arxiv updated 2019/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.