Collet, Eckmann and the bifurcation measure

The moduli space M_d of degree d≥ 2 rational maps can naturally be endowed with a measure μbif detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure μbif 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 μbif 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.