Speiser meets Misiurewicz

TL;DR

This paper introduces a Misiurewicz condition for transcendental entire functions, explores perturbations of Speiser functions, and demonstrates approximation by hyperbolic maps and measure-theoretic properties of these functions.

Contribution

It defines a new Misiurewicz condition for transcendental functions and analyzes the approximation and measure properties of Speiser functions satisfying this condition.

Findings

Every Misiurewicz entire function can be approximated by hyperbolic maps.

Misiurewicz functions are Lebesgue density points of hyperbolic maps if their Julia sets have zero measure.

The set of Misiurewicz Speiser functions has Lebesgue measure zero in parameter space.

Abstract

We propose a notion of Misiurewicz condition for transcendental entire functions and study perturbations of Speiser functions satisfying this condition in their parameter spaces (in the sense of Eremenko and Lyubich). We show that every Misiurewicz entire function can be approximated by hyperbolic maps in the same parameter space. Moreover, Misiurewicz functions are Lebesgue density points of hyperbolic maps if their Julia sets have zero Lebesgue measure. We also prove that the set of Misiurewicz Speiser functions has Lebesgue measure zero in the parameter space.