Hyperbolic components of rational maps: Quantitative equidistribution and counting

Let Lambda be a quasi-projective variety and assume that, either Lambda is a subvariety of the moduli space M-d of degree d rational maps, or Lambda parametrizes an algebraic family (f lambda) lambda is an element of Lambda of degree d rational maps on P-1. We prove the equidistribution of parameters having p distinct neutral cycles towards the bifurcation current T-bif (p) letting the periods of the cycles go to infinity, with an exponential speed of convergence. Several consequences of this result are: - a precise asymptotic of the number of hyperbolic components of parameters admitting 2d - 2 distinct attracting cycles of exact periods n(1), ..., n(2d -2 )as min(j) n(j) -> infinity in term of the mass of the bifurcation measure and compute that mass in the case where d = 2. In particular, in M-d, the number of such components is asymptotic to d(,)(n1+...+n2d-2) provided that min(j) n(j ) is large enough. - in the moduli space P-d of polynomials of degree d, among hyperbolic components such that all (finite) critical points are in the immediate basins of (not necessarily distinct) attracting cycles of respective exact periods n(1), ..., n(d-1), the proportion of those components, counted with multiplicity, having at least two critical points in the same basin of attraction is exponentially small. - in M-d, we prove the equidistribution of the centers of the hyperbolic components admitting 2d - 2 distinct attracting cycles of exact periods n(1), ...,n(2d-2) towards the bifurcation measure mu(bif) with an exponential speed of convergence. - we have equidistribution, up to extraction, of the parameters having p distinct cycles of given multipliers towards the bifurcation current T-bif (p) outside a pluripolar set of multipliers as the minimum of the periods of the cycles goes to infinity. As a by-product, we also get the weak genericity of hyperbolic postcritically finiteness in the moduli space of rational maps. A key step of the proof is a locally uniform version of the quantitative approximation of the Lyapunov exponent of a rational map by the log+ of the moduli of the multipliers of periodic points.