[Sommation de Borel et écart des séparatrices de l'application de Hénon]
L’application de Hénon est une transformation symplectique de possédant un point fixe parabolique auquel sont associées deux variétés invariantes complexes (les séparatrices). Une unique série formelle correspond à ces deux variétés, et nous étudions sa transformée de Borel formelle. Nous prouvons qu’elle définit un germe analytique et étudions sa surface de Riemann, ainsi que les singularités de son prolongement analytique. Nous donnons en particulier une description complète de la “première singularité”, et démontrons qu’une certaine constante qui détermine l’écart des séparatrices n’est pas nulle. Ces résultats sont aussi présentés dans le langage de la théorie de la résurgence.
We study two complex invariant manifolds associated with the para\-bolic 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.
Keywords: Hénon map, difference equations, splitting of separatrices, Borel summation, Laplace transform, resurgence
Mot clés : application de Hénon, équations aux différences, écart des séparatrices, sommation de Borel, transformation de Laplace, résurgence
@article{AIF_2001__51_2_513_0,
author = {Gelfreich, Vassili and Sauzin, David},
title = {Borel summation and splitting of separatrices for the {H\'enon} map},
journal = {Annales de l'Institut Fourier},
pages = {513--567},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {51},
number = {2},
year = {2001},
doi = {10.5802/aif.1831},
mrnumber = {1824963},
zbl = {0988.37031},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.1831/}
}
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 SP - 513 EP - 567 VL - 51 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1831/ DO - 10.5802/aif.1831 LA - en ID - AIF_2001__51_2_513_0 ER -
%0 Journal Article %A Gelfreich, Vassili %A Sauzin, David %T Borel summation and splitting of separatrices for the Hénon map %J Annales de l'Institut Fourier %D 2001 %P 513-567 %V 51 %N 2 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1831/ %R 10.5802/aif.1831 %G en %F AIF_2001__51_2_513_0
Gelfreich, Vassili; Sauzin, David. Borel summation and splitting of separatrices for the Hénon map. Annales de l'Institut Fourier, Tome 51 (2001) no. 2, pp. 513-567. doi : 10.5802/aif.1831. http://www.numdam.org/articles/10.5802/aif.1831/
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