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

Published online by Cambridge University Press:  08 November 2011

JÖRN PETER
JÖRN PETER*
Affiliation:
Mathematisches Seminar der Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Straße 4, 24098 Kiel, Germany (email: peter@math.uni-kiel.de)
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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 hγ(t)=t2g(1/t)γ, where g is the inverse of a linearizer of some exponential map and γ≥(log ρ(f)+K1)/c, but for ρ large enough, there exists a function fρ of bounded type with order ρ such that the Hausdorff measures of the escaping set and the Julia set of fρ with respect to hγ′ are zero whenever γ′ ≤(log ρ−K2)/c.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 1 , February 2013 , pp. 284 - 302
Copyright
Copyright © Cambridge University Press 2011

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