Higher bifurcation currents, neutral cycles and the Mandelbrot set

We prove that given any θ1, . . . , θ2d−2 ∈ R\Z, the support of the bifurcation measure of the moduli space of degree d rational maps coincides with the closure of classes of maps having 2d − 2 neutral cycles of respective multipliers e1 , . . . , e2d−2 . To this end, we generalize a famous result of McMullen, proving that homeomorphic copies of (∂M) are dense in the support of the k-bifurcation current T k bif in general families of rational maps, where M is the Mandelbrot set. As a consequence, we also get sharp dimension estimates for the supports of the bifurcation currents in any family.