Strong probabilistic stability in holomorphic families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$ and polynomial-like maps

TL;DR

This paper demonstrates that in stable families of endomorphisms of complex projective space, certain invariant measures can be holomorphically tracked across parameters, extending previous stability results to a broader class of measures and polynomial-like maps.

Contribution

It generalizes probabilistic stability results to measures with high entropy and positive Lyapunov exponents, providing new tools for understanding stability in complex dynamics.

Findings

Invariant measures with entropy > (k-1)log d can be holomorphically followed across parameters.

Almost all points in the Julia set can be tracked holomorphically without intersection.

Sufficient conditions for positivity of Lyapunov exponents in polynomial-like maps are established.

Abstract

We prove that, in stable families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$, all invariant measures whose measure-theoretic entropy is strictly larger than $(k-1)\log d$ at a given parameter can be followed holomorphically with the parameter in all the parameter space. As a consequence, almost all points (with respect to any such measure at any parameter) in the Julia set can be followed holomorphically without intersections. This generalizes previous results by Berteloot, Dupont, and the first author for the measure of maximal entropy, and provides a parallel in this setting to the probabilistic stability of H\'enon maps by Berger-Dujardin-Lyubich. Our proof relies both on techniques from the theory of stability/bifurcation in any dimension and on an explicit lower bound for the Lyapunov exponents for an ergodic measure in terms of its measure-theoretic entropy, due to de Th\'elin and Dupont. A local version of our result holds also for all measures supported on the Julia set with just strictly positive Lyapunov exponents and not charging the post-critical set. Analogous results hold in families of polynomial-like maps of large topological degree. In this case, as part of our proof, we also give a sufficient condition for the positivity of the Lyapunov exponents of an ergodic measure for a polynomial-like map in any dimension in term of its measure-theoretic entropy, generalizing to this setting the analogous result by de Th\'elin and Dupont valid on $\mathbb{P}^k(\mathbb{C})$.