Slowly recurrent Collet-Eckmann maps on the Riemann sphere

TL;DR

This paper investigates how rational Collet-Eckmann maps with slowly recurrent critical sets on the Riemann sphere behave under perturbations, showing they are Lebesgue density points within this class.

Contribution

It demonstrates that such maps are Lebesgue density points of Collet-Eckmann maps in the space of rational maps of the same degree.

Findings

Any such map is a Lebesgue density point of Collet-Eckmann maps.

The Julia set is the entire sphere for these maps.

Critical set recurrence is allowed to be slow.

Abstract

In this paper we study perturbations of rational Collet-Eckmann maps for which the Julia set is the whole sphere, and for which the critical set is allowed to be slowly recurrent. We show that any such map is a Lebesgue density point of Collet-Eckmann maps in the space of rational maps of the same degree at least $2$.