Lebesgue measure of Julia sets and escaping sets of certain entire functions

TL;DR

This paper establishes criteria under which the Julia set and escaping set of certain entire functions have positive Lebesgue measure, with applications to Poincaré functions of semihyperbolic polynomials and the Weierstrass sigma-function.

Contribution

It introduces new criteria for positive measure of Julia and escaping sets, extending understanding of their size for specific classes of entire functions.

Findings

Criteria for positive measure of Julia and escaping sets.

Application to Poincaré functions of semihyperbolic polynomials.

Application to the Weierstrass sigma-function.

Abstract

We give criteria for the escaping set and the Julia set of an entire function to have positive measure. The results are applied to Poincar\'e functions of semihyperbolic polynomials and to the Weierstra{\ss} $\sigma$-function.