A KAM phenomenon for singular holomorphic vector fields

Laurent Stolovitch

Publications Mathématiques de l'IHÉS (2005)

  • Volume: 102, page 99-165
  • ISSN: 0073-8301

Abstract

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Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form “resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining the invariant sets is of positive measure.

How to cite

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Stolovitch, Laurent. "A KAM phenomenon for singular holomorphic vector fields." Publications Mathématiques de l'IHÉS 102 (2005): 99-165. <http://eudml.org/doc/104215>.

@article{Stolovitch2005,
abstract = {Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form “resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining the invariant sets is of positive measure.},
author = {Stolovitch, Laurent},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {Hamiltonian systems; KAM theory; holomorphic vector field; (formal) normal form; (formal) first integral; resonant monomial; holomorphic invariant manifold; persistence},
language = {eng},
pages = {99-165},
publisher = {Springer},
title = {A KAM phenomenon for singular holomorphic vector fields},
url = {http://eudml.org/doc/104215},
volume = {102},
year = {2005},
}

TY - JOUR
AU - Stolovitch, Laurent
TI - A KAM phenomenon for singular holomorphic vector fields
JO - Publications Mathématiques de l'IHÉS
PY - 2005
PB - Springer
VL - 102
SP - 99
EP - 165
AB - Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection of a polydisc with an analytic set of the form “resonant monomials = constants”. Such a biholomorphism conjugates the restriction of X to one of its invariant varieties to the restriction of a linear diagonal vector field to a toric variety. Moreover, we show that the set of “frequencies” defining the invariant sets is of positive measure.
LA - eng
KW - Hamiltonian systems; KAM theory; holomorphic vector field; (formal) normal form; (formal) first integral; resonant monomial; holomorphic invariant manifold; persistence
UR - http://eudml.org/doc/104215
ER -

References

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