Normalization of bundle holomorphic contractions and applications to dynamics

Berteloot, François; Dupont, Christophe; Molino, Laura

doi:10.5802/aif.2409

Normalization of bundle holomorphic contractions and applications to dynamics
[Normalisation de contractions holomorphes fibrées et applications en dynamique]

Berteloot, François ¹ ; Dupont, Christophe ² ; Molino, Laura ³

¹ Université Toulouse III Institut Mathématique de Toulouse Équipe Émile Picard, Bat. 1R2 118, route de Narbonne 31062 Toulouse Cedex 9 (France)
² Université Paris XI-Orsay CNRS UMR 8628 Mathématique, Bât. 425 91405 Orsay Cedex (France)
³ Università di Parma Dipartimento di Matematica Parco Area delle Scienze, Viale Usberti 53/A 43100 Parma (Italia)

Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2137-2168.

Résumé
Abstract

Nous démontrons un théorème de Poincaré-Dulac pour des suites de contractions holomorphes ${(G_{n})}_{n \in ℤ}$ à différentielles $d_{0} G_{n}$ scindées. Les relations de résonance qui déterminent les formes normales portent sur les modules des taux exponentiels de contractions. Les résultats sont formulés dans le cadre des applications fibrées.

De telles suites de contractions holomorphes apparaissent naturellement comme branches inverses d’endomorphismes de $ℂ ℙ^{k}$ . Dans ce contexte, notre résultat de normalisation nous permet d’estimer précisément les distorsions des ellipsoides le long d’orbites typiques. Nous en déduisons que les exposants de Lyapounov de la mesure d’équilibre sont approchés par les multiplicateurs des cycles répulsifs.

We establish a Poincaré-Dulac theorem for sequences ${(G_{n})}_{n \in ℤ}$ of holomorphic contractions whose differentials $d_{0} G_{n}$ split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of contraction. Our results are actually stated in the framework of bundle maps.

Such sequences of holomorphic contractions appear naturally as iterated inverse branches of endomorphisms of $ℂ ℙ^{k}$ . In this context, our normalization result allows to estimate precisely the distortions of ellipsoids along typical orbits. As an application, we show how the Lyapunov exponents of the equilibrium measure are approximated in terms of the multipliers of the repulsive cycles.

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/aif.2409

Classification : 37F10, 37G05, 32H50
Keywords: Normalization, Poincaré-Dulac theorem, Lyapounov exponents
Mot clés : Normalisation, théorème de Poincaré-Dulac, exposants de Lyapounov

Affiliations des auteurs :

Berteloot, François ¹ ; Dupont, Christophe ² ; Molino, Laura ³

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Berteloot, François; Dupont, Christophe; Molino, Laura. Normalization of bundle holomorphic contractions and applications to dynamics. Annales de l'Institut Fourier, Tome 58 (2008) no. 6, pp. 2137-2168. doi : 10.5802/aif.2409. https://www.numdam.org/articles/10.5802/aif.2409/

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