The bifurcation measure has maximal entropy

Let A be a complex manifold and let (f lambda)lambda is an element of?\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${({f_\lambda})_{\lambda \in {\rm{\Lambda}}}}$$\end{document} be a holomorphic family of rational maps of degree d >= 2of DOUBLE-STRUCK CAPITAL P-1. We define a natural notion of entropy of bifurcation, mimicking the classical definition of entropy, by the parametric growth rate of critical orbits. We also define a notion of a measure-theoretic bifurcation entropy for which we prove a variational principle: the measure of bifurcation is a measure of maximal entropy. We rely crucially on a generalization of Yomdin's bound of the volume of the image of a dynamical ball. Applying our results to complex dynamics in several variables, we notably define and compute the entropy of the trace measure of the Green currents of a holomorphic endomorphism of DOUBLE-STRUCK CAPITAL P-k.