Collet, Eckmann and the bifurcation measure

Abstract

The moduli space ${\mathcal{M}}_d$ of degree $d\ge 2$ 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.