The bifurcation measure has maximal entropy

TL;DR

This paper introduces a new entropy measure for bifurcations in complex dynamics, proves a variational principle linking it to measures of maximal entropy, and extends the concept to higher-dimensional complex systems.

Contribution

It defines a novel bifurcation entropy, establishes a variational principle, and generalizes the concept to complex endomorphisms of projective spaces.

Findings

Bifurcation entropy is maximized by the measure of bifurcation.

A variational principle relates bifurcation entropy to measure-theoretic entropy.

The method extends to higher-dimensional complex dynamics, including Green currents.

Abstract

Let $\Lambda$ be a complex manifold and let $(f_\lambda)_{\lambda\in \Lambda}$ be a holomorphic family of rational maps of degree $d\geq 2$ of $\mathbb{P}^1$. 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 $\mathbb{P}^k$.