Hausdorff measure of escaping and Julia sets for bounded type functions of finite order

2011/02/24 by Jörn Peter, Peter, Jörn
Mathematics · #30D05 (Primary) 37F10 (Secondary) #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:30D05 #msc:37F10

paper · pdf · doi:10.48550/arxiv.1102.4933

21 pages

arxiv created 2011/02/24 · arxiv updated 2011/02/25

Abstract

We show that the escaping sets and the Julia sets of bounded type transcendental entire functions of order ρ become 'smaller' as ρ→∞. More precisely, their Hausdorff measures are infinite with respect to the gauge function h_γ(t)=t²g(1/t)^γ, where g is the inverse of a linearizer of some exponential map and γ≥(logρ(f)+K₁)/c, but for ρ large enough, there exists a function f_ρ of bounded type with order ρ such that the Hausdorff measures of the escaping set and the Julia set of f_ρ with respect to h_γ' are zero whenever γ'≤(logρ-K₂)/c.

Related

• Hausdorff measure of escaping sets on certain meromorphic functions
• Hausdorff dimension of elliptic functions with critical values approaching infinity
• Hausdorff dimension of Julia sets in the logistic family
• A generalized family of transcendental functions with one dimensional Julia sets
• Hausdorff dimension and conformal measures of Feigenbaum Julia sets