Title :

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

Author(s) :

Bianchi, Fabrizio [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Centre National de la Recherche Scientifique [CNRS]
Rakhimov, Karim [Auteur]
National University of Singapore [NUS]

English keyword(s) :

Holomorphic families
Stability
Polynomial-like maps
Lyapunov exponents

HAL domain(s) :

Mathématiques [math]/Systèmes dynamiques [math.DS]
Mathématiques [math]/Variables complexes [math.CV]

English abstract : [en]

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 ...
Show more >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 Hénon 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 Thélin 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 Thélin and Dupont valid on $\mathbb P^k (\mathbb C)$.Show less >

Language :

Anglais

ANR Project :

ULNE
Centre Européen pour les Mathématiques, la Physique et leurs Interactions
Quantisation, singularités et dynamique holomorphe
Dynamique parabolique, bifurcations et domaines errants

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL