Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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F. Bianchi - K. Rakhimov

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

created by bianchi on 09 Jul 2022
modified on 04 Jul 2024


Accepted Paper

Inserted: 9 jul 2022
Last Updated: 4 jul 2024

Journal: International Mathematics Research Notices (IMRN)
Year: 2022

ArXiv: 2206.04165 PDF


We prove that, in stable families of endomorphisms of $\mathbb{P}^k(\mathbb{C})$, all invariant measures with strictly positive Lyapunov exponents at a given parameter can be followed holomorphically with the parameter. 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. An analogous result holds for families of polynomial-like maps whose topological degree dominates the other dynamical degrees. This generalizes previous results for the measure of maximal entropy. We also prove a sufficient condition for the positivity of the Lyapunov exponents of a measure in term of the metric entropy, generalizing an analogous result by de Th\'elin and Dupont for the endomorphisms of $\mathbb{P}^k(\mathbb{C})$ to the setting of polynomial-like map. Our work provides a parallel in this setting to the probabilistic stability of H\'enon maps by Berger-Dujardin-Lyubich.

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