Strong Probabilistic Stability in Holomorphic Families of Endomorphisms of ℙk (ℂ) and Polynomial-Like Maps

We prove that, in stable families of endomorphisms of $\mathbb{P}^{k} (\mathbb C)$, the measurable holomorphic motion of the Julia sets introduced by Berteloot, Dupont, and the first author is unbranched at almost every point with respect to all measures on the Julia set with strictly positive Lyapunov exponents and not charging the post-critical set. This provides a parallel in this setting to the probabilistic stability of Hénon maps by Berger–Dujardin–Lyubich. An analogous result holds in families of polynomial-like maps of large topological degree. In this case, we also give a sufficient condition for the positivity of the Lyapunov exponents of an ergodic measure in terms of its measure-theoretic entropy, generalizing to this setting an analogous result by de Thélin and Dupont valid on $\mathbb{P}^{k} (\mathbb C)$.