# Collet, Eckmann and the bifurcation measure

## TL;DR

This paper proves that the bifurcation measure in the moduli space of degree d rational maps has a support of positive Lebesgue measure, by identifying a large set of Collet-Eckmann maps satisfying certain conditions.

## Contribution

It establishes a general condition for rational maps to be in the support of the bifurcation measure and shows a positive measure set of Collet-Eckmann maps meet this condition.

## Key findings

- Support of bifurcation measure has positive Lebesgue measure.
- Large set of Collet-Eckmann maps satisfy the support condition.
- Existence of positive measure of maps approximated by hyperbolic maps.

## Abstract

The moduli space $\mathcal{M}_d$ of degree $d\geq2$ rational maps can naturally be endowed with a measure $\mu_\mathrm{bif}$ detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure $\mu_\mathrm{bif}$ has positive Lebesgue measure. To do so, we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of $\mu_\mathrm{bif}$ and we exhibit a large set of Collet-Eckmann rational maps which satisfy this condition. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps.