# zbMATH — the first resource for mathematics

Normalization of bundle holomorphic contractions and applications to dynamics. (English) Zbl 1151.37038
Summary: We establish a Poincaré-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.

##### MSC:
 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 37G05 Normal forms for dynamical systems 32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
Full Text:
##### References:
 [1] Bedford, E.; Lyubich, M.; Smillie, J., Distribution of periodic points of polynomial diffeomorphisms of $$\mathbf{C}^2,$$ Invent. Math., 114, 2, 277-288, (1993) · Zbl 0799.58039 [2] Berteloot, F., Méthodes de changement d’échelles en analyse complexe, Ann. Fac. Sci. Toulouse Math. (6), 15, 3, 427-483, (2006) · Zbl 1123.37019 [3] Berteloot, F.; Bassanelli, G., Bifurcation currents in holomorphic dynamics in $$\mathbb{P}^k,$$ J. Reine Angew. Math., 608, 201-235, (2007) · Zbl 1136.37025 [4] Berteloot, F.; Dupont, C., Une caractérisation des endomorphismes de Lattès par leur mesure de Green, Comment. Math. Helv., 80, 2, 433-454, (2005) · Zbl 1079.37039 [5] Briend, J. Y.; Duval, J., Exposants de liapounoff et distribution des points périodiques d’un endomorphisme de $$\mathbb{C}\mathbb{P}^k,$$ Acta Math., 182, 2, 143-157, (1999) · Zbl 1144.37436 [6] Dinh, T. C.; Sibony, N., Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9), 82, 4, 367-423, (2003) · Zbl 1033.37023 [7] Fornæss, J. E.; Stensønes, B., Stable manifolds of holomorphic hyperbolic maps, Internat. J. Math., 15, 8, 749-758, (2004) · Zbl 1071.32016 [8] Guysinsky, M.; Katok, A., Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations, Math. Res. Lett., 5, 1-2, 149-163, (1998) · Zbl 0988.37063 [9] Jonsson, M.; Varolin, D., Stable manifolds of holomorphic diffeomorphisms, Invent. Math., 149, 2, 409-430, (2002) · Zbl 1048.37047 [10] Katok, A.; Hasselblatt, B., Introduction to the modern theory of dynamical systems, (1995), Cambridge Univ. Press · Zbl 0878.58020 [11] Katok, A.; Spatzier, R., Nonstationary normal forms and rigidity of group actions, Electron. Res. Announc. Amer. Math. Soc., 2, 3, 124-133, (1996) · Zbl 0871.58073 [12] Peters, H., Perturbed basins of attraction, Math. Ann., 337, 1, 1-13, (2007) · Zbl 1112.37037 [13] Rosay, J. P.; Rudin, W., Holomorphic maps from $$\mathbb{C}^n$$ to $$\mathbb{C}^n,$$ Trans. Amer. Math. Soc., 310, 1, 47-86, (1988) · Zbl 0708.58003 [14] Sibony, N., Dynamique et Géométrie Complexes, Panoramas et Synthèses, 8, Dynamique des applications rationnelles de $$\mathbb{P}^k, (1999),$$ SMF et EDP Sciences · Zbl 1020.37026 [15] Sternberg, S., Local contractions and a theorem by Poincaré, Am. J. Math., 79, 809-824, (1957) · Zbl 0080.29902 [16] Szpiro, L.; Tucker, T. J., Equidistribution and generalized Mahler measure, (2005) · Zbl 1283.37075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.