Slowly recurrent Collet-Eckmann maps with non-empty Fatou set

TL;DR

This paper investigates rational Collet-Eckmann maps with non-constant Julia sets and slowly recurrent critical points, showing they are Lebesgue density points of hyperbolic maps under certain conditions.

Contribution

It proves that such maps are Lebesgue density points of hyperbolic maps in specific families, extending understanding of their stability and recurrence properties.

Findings

Maps with fixed critical point orders are Lebesgue density points of hyperbolic maps.

If all critical points are simple, they are density points in the full space of rational maps.

The results apply to maps with Julia sets not equal to the entire sphere.

Abstract

In this paper we study rational Collet-Eckmann maps for which the Julia set is not the whole sphere and for which the critical points are recurrent at a slow rate. In families where the orders of the critical points are fixed, we prove that such maps are Lebesgue density points of hyperbolic maps. In particular, if all critical points are simple, they are Lebesgue density points of hyperbolic maps in the full space of rational maps of any degree $d \geq 2$.