Entire functions with Julia sets of positive measure

TL;DR

This paper investigates entire functions with bounded critical and asymptotic values, establishing conditions under which their Julia sets have positive measure, and highlighting the role of function order in this property.

Contribution

It provides new criteria linking the growth rate of entire functions to the measure of their Julia sets, refining previous estimates with a sharpened Tsuji estimate.

Findings

Julia sets have positive measure when the function's growth is controlled.

Order exceeding N/2 can lead to Julia sets of zero measure.

A refined Tsuji estimate is used to establish these results.

Abstract

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f(z)|>R has N components for some R>0, then the order of f is at least N/2. More precisely, we have log log M(r,f) > (N/2) log r - O(1), where M(r,f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Tsuji related to the Denjoy-Carleman-Ahlfors theorem.