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

TL;DR

This paper investigates how the Hausdorff measure of escaping and Julia sets for bounded type transcendental entire functions varies with the function's order, revealing that these measures tend to zero for large order under certain conditions.

Contribution

It provides a detailed analysis of the Hausdorff measure of Julia and escaping sets for bounded type functions, showing how these measures diminish as the order increases, and constructs specific functions illustrating this behavior.

Findings

Hausdorff measures are infinite for certain gauge functions at finite order

Existence of functions with zero Hausdorff measure for large order

Measures decrease as the order of the function increases

Abstract

We show that the escaping sets and the Julia sets of bounded type transcendental entire functions of order $\rho$ become 'smaller' as $\rho\to\infty$. More precisely, their Hausdorff measures are infinite with respect to the gauge function $h_\gamma(t)=t^2g(1/t)^\gamma$, where $g$ is the inverse of a linearizer of some exponential map and $\gamma\geq(\log\rho(f)+K_1)/c$, but for $\rho$ large enough, there exists a function $f_\rho$ of bounded type with order $\rho$ such that the Hausdorff measures of the escaping set and the Julia set of $f_\rho$ with respect to $h_{\gamma'}$ are zero whenever $\gamma'\leq(\log\rho-K_2)/c$.