Borel summation and splitting of separatrices for the Hénon map

Vassili Gelfreich[1]; David Sauzin[2]

  • [1] The Steklov Mathematical Institute at St. Petersburg, St. Petersburg (Russie)
  • [2] Institut de Mécanique Céleste, Astronomie et systèmes Dynamiques, 77 avenue Denfert-Rochereau, 75014 Paris (France)

Annales de l’institut Fourier (2001)

  • Volume: 51, Issue: 2, page 513-567
  • ISSN: 0373-0956

Abstract

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We study two complex invariant manifolds associated with the parabolic fixed point of the area-preserving Hénon map. A single formal power series corresponds to both of them. The Borel transform of the formal series defines an analytic germ. We explore the Riemann surface and singularities of its analytic continuation. In particular we give a complete description of the “first” singularity and prove that a constant, which describes the splitting of the invariant manifolds, does not vanish. An interpretation in terms of Resurgence theory is also given.

How to cite

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Gelfreich, Vassili, and Sauzin, David. "Borel summation and splitting of separatrices for the Hénon map." Annales de l’institut Fourier 51.2 (2001): 513-567. <http://eudml.org/doc/115924>.

@article{Gelfreich2001,
abstract = {We study two complex invariant manifolds associated with the parabolic fixed point of the area-preserving Hénon map. A single formal power series corresponds to both of them. The Borel transform of the formal series defines an analytic germ. We explore the Riemann surface and singularities of its analytic continuation. In particular we give a complete description of the “first” singularity and prove that a constant, which describes the splitting of the invariant manifolds, does not vanish. An interpretation in terms of Resurgence theory is also given.},
affiliation = {The Steklov Mathematical Institute at St. Petersburg, St. Petersburg (Russie); Institut de Mécanique Céleste, Astronomie et systèmes Dynamiques, 77 avenue Denfert-Rochereau, 75014 Paris (France)},
author = {Gelfreich, Vassili, Sauzin, David},
journal = {Annales de l’institut Fourier},
keywords = {Hénon map; difference equations; splitting of separatrices; Borel summation; Laplace transform; resurgence; quadratic area-preserving map; divergent asymptotic series},
language = {eng},
number = {2},
pages = {513-567},
publisher = {Association des Annales de l'Institut Fourier},
title = {Borel summation and splitting of separatrices for the Hénon map},
url = {http://eudml.org/doc/115924},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Gelfreich, Vassili
AU - Sauzin, David
TI - Borel summation and splitting of separatrices for the Hénon map
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 2
SP - 513
EP - 567
AB - We study two complex invariant manifolds associated with the parabolic fixed point of the area-preserving Hénon map. A single formal power series corresponds to both of them. The Borel transform of the formal series defines an analytic germ. We explore the Riemann surface and singularities of its analytic continuation. In particular we give a complete description of the “first” singularity and prove that a constant, which describes the splitting of the invariant manifolds, does not vanish. An interpretation in terms of Resurgence theory is also given.
LA - eng
KW - Hénon map; difference equations; splitting of separatrices; Borel summation; Laplace transform; resurgence; quadratic area-preserving map; divergent asymptotic series
UR - http://eudml.org/doc/115924
ER -

References

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Citations in EuDML Documents

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  1. Yafei Ou, On the stability by convolution product of a resurgent algebra
  2. Jean-Marc Rasoamanana, Résurgence-sommabilité de séries formelles ramifiées dépendant d’un paramètre et solutions d’équations différentielles linéaires
  3. Carme Olivé, David Sauzin, Tere M. Seara, Resurgence in a Hamilton-Jacobi equation
  4. José Pedro Gaivão, Analytic invariants for the 1 : - 1 resonance
  5. Ovidiu Costin, Stavros Garoufalidis, Resurgence of the Kontsevich-Zagier series

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