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

Jörn Peter

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arXiv · 1102.4933

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

Jörn Peter

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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 $ρ\to\infty$. More precisely, their Hausdorff measures are infinite with respect to the gauge function $h_γ(t)=t^2g(1/t)^γ$, where $g$ is the inverse of a linearizer of some exponential map and $γ\geq(\logρ(f)+K_1)/c$, 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 $γ'\leq(\logρ-K_2)/c$.

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Jörn Peter. 2011-02-24. Hausdorff measure of escaping and Julia sets for bounded type functions of finite order. https://arxiv.org/abs/1102.4933

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