Normalization of bundle holomorphic contractions and applications to dynamics

Berteloot, François; Dupont, Christophe; Molino, Laura

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

Nous démontrons un théorème de Poincaré-Dulac pour des suites de contractions holomorphes ${\left(G_n\right)}_{n\in \mathbb{Z}}$ 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 $ℂ{\mathbb{P}}^k$. 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 ${\left(G_n\right)}_{n\in \mathbb{Z}}$ 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 $ℂ{\mathbb{P}}^k$. 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.

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