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The bifurcation measure has maximal entropy

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Abstract

Let A be a complex manifold and let \({({f_\lambda})_{\lambda \in {\rm{\Lambda}}}}\) be a holomorphic family of rational maps of degree d ≥ 2of ℙ1. We define a natural notion of entropy of bifurcation, mimicking the classical definition of entropy, by the parametric growth rate of critical orbits. We also define a notion of a measure-theoretic bifurcation entropy for which we prove a variational principle: the measure of bifurcation is a measure of maximal entropy. We rely crucially on a generalization of Yomdin’s bound of the volume of the image of a dynamical ball.

Applying our results to complex dynamics in several variables, we notably define and compute the entropy of the trace measure of the Green currents of a holomorphic endomorphism of ℙk.

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Authors and Affiliations

  1. Laboratoire d’Analyse, Géométrie et Applications, CNRS-UMR 7539, Institut Galilée, Université Paris 13, 99 avenue J. B. Clément, 93430, Villetaneuse, France

    Henry De Thélin

  2. Centre de Mathématiques Laurent Schwartz, École Polytechnique, Université Paris-Saclay, 91128, Palaiseau Cedex, France

    Thomas Gauthier

  3. Laboratoire Amiénois de Mathématique Fondamentale et Appliquée CNRS-UMR 7352, Université de Picardie Jules Verne, 33 rue Saint-Leu, 80039, AMIENS Cedex 1, France

    Gabriel Vigny

Authors
  1. Henry De Thélin
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  2. Thomas Gauthier
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  3. Gabriel Vigny
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Corresponding author

Correspondence to Gabriel Vigny.

Additional information

The second and third authors’ research is partially supported by the ANR grant Fatou ANR-17-CE40-0002-01.

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De Thélin, H., Gauthier, T. & Vigny, G. The bifurcation measure has maximal entropy. Isr. J. Math. 235, 213–243 (2020). https://doi.org/10.1007/s11856-019-1955-6

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  • DOI: https://doi.org/10.1007/s11856-019-1955-6

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