The bifurcation measure has maximal entropy

Abstract

Let A be a complex manifold and let $${({f_\lambda})_{\lambda \in {\rm{\Lambda}}}}$$ be a holomorphic family of rational maps of degree d ≥ 2of ℙ1. 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 ℙk.