Normalization of bundle holomorphic contractions and applications to dynamics

François Berteloot[1]; Christophe Dupont[2]; Laura Molino[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 (2008)

  • Volume: 58, Issue: 6, page 2137-2168
  • ISSN: 0373-0956

Abstract

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We establish a Poincaré-Dulac theorem for sequences ( G n ) n ∈ 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.

How to cite

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Berteloot, François, Dupont, Christophe, and Molino, Laura. "Normalization of bundle holomorphic contractions and applications to dynamics." Annales de l’institut Fourier 58.6 (2008): 2137-2168. <http://eudml.org/doc/10373>.

@article{Berteloot2008,
abstract = {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.},
affiliation = {Université Toulouse III Institut Mathématique de Toulouse Équipe Émile Picard, Bat. 1R2 118, route de Narbonne 31062 Toulouse Cedex 9 (France); Université Paris XI-Orsay CNRS UMR 8628 Mathématique, Bât. 425 91405 Orsay Cedex (France); Università di Parma Dipartimento di Matematica Parco Area delle Scienze, Viale Usberti 53/A 43100 Parma (Italia)},
author = {Berteloot, François, Dupont, Christophe, Molino, Laura},
journal = {Annales de l’institut Fourier},
keywords = {Normalization; Poincaré-Dulac theorem; Lyapounov exponents; normal forms; bundle maps},
language = {eng},
number = {6},
pages = {2137-2168},
publisher = {Association des Annales de l’institut Fourier},
title = {Normalization of bundle holomorphic contractions and applications to dynamics},
url = {http://eudml.org/doc/10373},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Berteloot, François
AU - Dupont, Christophe
AU - Molino, Laura
TI - Normalization of bundle holomorphic contractions and applications to dynamics
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 6
SP - 2137
EP - 2168
AB - 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.
LA - eng
KW - Normalization; Poincaré-Dulac theorem; Lyapounov exponents; normal forms; bundle maps
UR - http://eudml.org/doc/10373
ER -

References

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