an:05367572
Zbl 1151.37038
Berteloot, Fran??ois; Dupont, Christophe; Molino, Laura
Normalization of bundle holomorphic contractions and applications to dynamics
EN
Ann. Inst. Fourier 58, No. 6, 2137-2168 (2008).
00233543
2008
j
37F10 37G05 32H50
normal forms; Poincar??-Dulac theorem; Lyapounov exponents; bundle maps
Summary: We establish a Poincar??-Dulac theorem for sequences \(({G_{ n })_{ n\in \mathbb Z }}\) 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 \(\mathbb C\mathbb P^{ k }\). 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.