The bifurcation measure has maximal entropy

Abstract

Let Λ be a complex manifold and let (f λ ) λ∈Λ be a holomorphic family of rational maps of degree d ≥ 2 of 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 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 technics 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 P k .

Parametric Lyapunov exponents

“…As such, it is interesting to develop an ergodic theory for the bifurcation currents. This is what we did in [5] where we defined a notion of parametric entropy and proved, for example, that in a one‐dimensional family, the measure is a measure of maximal entropy. Pursuing our study, in the present article, we address the notion of parametric Lyapunov exponent.…”

Dynamical pairs with an absolutely continuous bifurcation measure

“…[2,3,11,12,13,18,19,20]. Note also that the entropy theory of dynamical pairs has been recently developed in [9].…”