Levi-flat invariant sets of holomorphic symplectic mappings

Xianghong Gong[1]

  • [1] University of Wisconsin-Madison, Department of Mathematics, Van Vleck Hall, 480 Lincoln Drive, Madison WI 53706-1388 (USA)

Annales de l’institut Fourier (2001)

  • Volume: 51, Issue: 1, page 151-208
  • ISSN: 0373-0956

Abstract

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We classify four families of Levi-flat sets which are defined by quadratic polynomials and invariant under certain linear holomorphic symplectic maps. The normalization of Levi- flat real analytic sets is studied through the technique of Segre varieties. The main purpose of this paper is to apply the Levi-flat sets to the study of convergence of Birkhoff's normalization for holomorphic symplectic maps. We also establish some relationships between Levi-flat invariant sets and first-integrals or meromorphic eigenfunctions of such maps. The results obtained for holomorphic symplectic maps are also applicable to holomorphic Hamiltonian systems via time-one maps.

How to cite

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Gong, Xianghong. "Levi-flat invariant sets of holomorphic symplectic mappings." Annales de l’institut Fourier 51.1 (2001): 151-208. <http://eudml.org/doc/115907>.

@article{Gong2001,
abstract = {We classify four families of Levi-flat sets which are defined by quadratic polynomials and invariant under certain linear holomorphic symplectic maps. The normalization of Levi- flat real analytic sets is studied through the technique of Segre varieties. The main purpose of this paper is to apply the Levi-flat sets to the study of convergence of Birkhoff's normalization for holomorphic symplectic maps. We also establish some relationships between Levi-flat invariant sets and first-integrals or meromorphic eigenfunctions of such maps. The results obtained for holomorphic symplectic maps are also applicable to holomorphic Hamiltonian systems via time-one maps.},
affiliation = {University of Wisconsin-Madison, Department of Mathematics, Van Vleck Hall, 480 Lincoln Drive, Madison WI 53706-1388 (USA)},
author = {Gong, Xianghong},
journal = {Annales de l’institut Fourier},
keywords = {Levi-flat set; Segre variety; holomorphic symplectic map; Birkhoff normal form; Levi-flat},
language = {eng},
number = {1},
pages = {151-208},
publisher = {Association des Annales de l'Institut Fourier},
title = {Levi-flat invariant sets of holomorphic symplectic mappings},
url = {http://eudml.org/doc/115907},
volume = {51},
year = {2001},
}

TY - JOUR
AU - Gong, Xianghong
TI - Levi-flat invariant sets of holomorphic symplectic mappings
JO - Annales de l’institut Fourier
PY - 2001
PB - Association des Annales de l'Institut Fourier
VL - 51
IS - 1
SP - 151
EP - 208
AB - We classify four families of Levi-flat sets which are defined by quadratic polynomials and invariant under certain linear holomorphic symplectic maps. The normalization of Levi- flat real analytic sets is studied through the technique of Segre varieties. The main purpose of this paper is to apply the Levi-flat sets to the study of convergence of Birkhoff's normalization for holomorphic symplectic maps. We also establish some relationships between Levi-flat invariant sets and first-integrals or meromorphic eigenfunctions of such maps. The results obtained for holomorphic symplectic maps are also applicable to holomorphic Hamiltonian systems via time-one maps.
LA - eng
KW - Levi-flat set; Segre variety; holomorphic symplectic map; Birkhoff normal form; Levi-flat
UR - http://eudml.org/doc/115907
ER -

References

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  1. M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277-291 Zbl0172.05301MR232018
  2. E. Bedford, Holomorphic continuation of smooth functions over Levi-flat hypersurfaces, Trans. Amer. Math. Soc. 232 (1977), 323-341 Zbl0382.32009MR481100
  3. G.D. Birkhoff, Surface transformations and their dynamical applications, Acta Math. 43 (1920), 1-119 Zbl47.0985.03
  4. G.D. Birkhoff, Dynamical Systems, Coll. Publ. 1927 vol. 9 (1966 (reprinted)), AMS Zbl0171.05402
  5. F. Bruhat, H. Cartan, Sur la structure des sous-ensembles analytiques réels, C. R. Acad. Sci. Paris 244 (1957), 988-991 Zbl0081.17201MR86108
  6. D. Burns, X. Gong, Singular Levi-flat real analytic hypersurfaces, Amer. J. Math. 121 (1999), 23-53 Zbl0931.32009MR1704996
  7. H. Cartan, Variétés analytiques réelles et variétés analytiques complexes, Bull. Soc. Math. France 85 (1957), 77-99 Zbl0083.30502MR94830
  8. K. Diederich, J.E. Fornaess, Pseudoconvex domains with real-analytic boundary, Ann. Math. (2) 107 (1978), 371-384 Zbl0378.32014MR477153
  9. L.H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals--elliptic case, Comment. Math. Helv. 65 (1990), 4-35 Zbl0702.58024MR1036125
  10. H. Ito, Convergence of Birkhoff normal forms for integrable systems, Comment. Math. Helv. 64 (1989), 412-461 Zbl0686.58021MR998858
  11. J.K. Moser, The analytic invariants of an area-preserving mapping near a hyperbolic fixed point, Comm. Pure Appl. Math. 9 (1956), 673-692 Zbl0072.40801MR86981
  12. H. Rüssmann, Über die Existenz einer Normalform inhaltstreuer elliptischer Transformationen, Math. Ann. 137 (1959), 64-77 Zbl0084.19904MR100064
  13. H. Rüssmann, Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann. 154 (1964), 285-300 Zbl0124.04701MR179409
  14. H. Rüssmann, Über die Normalform analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann. 169 (1967), 55-72 Zbl0143.10801MR213679
  15. B. Segre, Intorno al problem di Poincaré della rappresentazione pseudo-conform, Rend. Acc. Lincei 13 (1931), 676-683 Zbl0003.21302
  16. C.L. Siegel, On integrals of canonical systems, Ann. Math. 42 (1941), 806-822 Zbl0025.26503MR5818
  17. C.L. Siegel, Über die Existenz einer Normalform analytischer Hamiltonscher Differntialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann. 128 (1954), 144-170 Zbl0057.32002MR67298
  18. J. Vey, Sur certains systèmes dynamiques séparables, Amer. J. Math. 100 (1978), 591-614 Zbl0384.58012MR501141
  19. S.M. Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math. 43 (1977), 53-68 Zbl0348.32005MR463482

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