Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of ℙk
[Stabilité dynamique et exposants de Lyapunov pour les endomorphismes holomorphes de ℙk ]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 1, pp. 215-262.

Nous introduisons une notion de stabilité pour les mesures d'équilibre des familles holomorphes d'endomorphismes de ℙk et démontrons qu'elle est équivalente à la stabilité des cycles répulsifs et équivalente à l'existence d'un mouvement holomorphe mesurable des ensembles de Julia, appelé lamination d'équilibre. Nous caractérisons les bifurcations correspondantes par la sous-harmonicité stricte de la somme des exposants de Lyapunov ou par l'instabilité de la dynamique critique, nous analysons aussi comment les cycles répulsifs peuvent bifurquer. Nos méthodes reposent sur les propriétés des exposants de Lyapunov, sur la théorie ergodique et sur la théorie du pluripotentiel.

We introduce a notion of stability for equilibrium measures in holomorphic families of endomorphisms of ℙk and prove that it is equivalent to the stability of repelling cycles and equivalent to the existence of some measurable holomorphic motion of Julia sets which we call equilibrium lamination. We characterize the corresponding bifurcations by the strict subharmonicity of the sum of Lyapunov exponents or the instability of critical dynamics and analyze how repelling cycles may bifurcate. Our methods deeply exploit the properties of Lyapunov exponents and are based on ergodic and pluripotential theory.

Publié le :
DOI : 10.24033/asens.2355
Classification : 32H50, 32U40, 37F45, 37F50, 37H15.
Keywords: Holomorphic dynamics, dynamical stability, positive currents, Lyapunov exponents.
Mot clés : Dynamique holomorphe, stabilité dynamique, courants positifs, exposants de Lyapunov. 
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     author = {Berteloot, Fran\c{c}ois and Bianchi, Fabrizio and Dupont, Christophe},
     title = {Dynamical stability and {Lyapunov} exponents for holomorphic endomorphisms of~$\mathbb {P}^k$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {215--262},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
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Berteloot, François; Bianchi, Fabrizio; Dupont, Christophe. Dynamical stability and Lyapunov exponents for holomorphic endomorphisms of $\mathbb {P}^k$. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 1, pp. 215-262. doi : 10.24033/asens.2355. http://www.numdam.org/articles/10.24033/asens.2355/

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