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

Author:

Bianchi Fabrizio1,Rakhimov Karim23

Affiliation:

1. Dipartimento di Matematica, Università di Pisa , Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

2. V.I.Romanovskiy Institute of Mathematics of Uzbek Academy of Sciences , 100174 Tashkent, Uzbekistan

3. National University of Uzbekistan , 100174 Tashkent, Uzbekistan

Abstract

Abstract 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 $\mathbb{P}^{k} (\mathbb C)$.

Publisher

Oxford University Press (OUP)

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Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. A Mañé-Manning formula for expanding measures for endomorphisms of ℙ^{};Transactions of the American Mathematical Society;2024-09-05

2. Monotonicity of dynamical degrees for Hénon-like and polynomial-like maps;Transactions of the American Mathematical Society;2024-06-28

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