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A KAM phenomenon for singular holomorphic vector fields

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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.

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References

  1. V. I. Arnold, The stability of the equlibrium position of a hamiltonian system of ordinary differential equations in the general elliptique case, Soviet Math. Dokl., 2 (1961), 247–249.

    Google Scholar 

  2. V. I. Arnold, Proof of a theorem by A. N. Kolmogorov on the persistence of quasi-periodic motions under small perturbations of the hamiltonian, Russ. Math. Surv., 18 (1963), 9–36.

    Google Scholar 

  3. V. I. Arnold, Small denominators and the problem of stability of motion in the classical and celestian mechanics, Russ. Math. Surv., 18 (1963), 85–191.

    Google Scholar 

  4. V. I. Arnold, Méthodes mathématiques de la mécanique classique, Mir, 1976.

  5. V. I. Arnold, Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Mir, 1980.

  6. V. I. Arnold (ed.), Dynamical systems III, vol. 28 of Encyclopaedia of Mathematical Sciences, Springer, 1988.

  7. V. I. Bakhtin, A strengthened extremal property of Chebyshev polynomials, Moscow Univ. Math. Bull., 42 (1987), 24–26.

  8. V. I. Bernik and M. M. Dodson, Metric diophantine approximation on manifolds, vol. 137 of Cambridge Tracts in Mathematics, Cambridge University Press, 1999.

  9. H. W. Broer, G. W. Huitema, and M. B. Sevryuk, Quasi-periodic motions in famillies of dynamical systems, Lect. Notes Math.1645, Springer, 1996.

  10. Yu. I. Bibikov, Local theory of nonlinear analytic ordinary differential equations, Lect. Notes Math.702, Springer, 1979.

  11. J.-B. Bost, Tores invariants des systèmes dynamiques hamiltoniens (d’après Kolomogorov, Arnol’d, Moser, Rüssmann, Zehnder, Herman, Pöschel, ...), in Séminaire Bourbaki, Astérisque, 133–134 (1986), 113–157, Société Mathématiques de France, exposé 639.

  12. Yu. I. Bibikov and V. A. Pliss, On the existence of invariant tori in a neighbourhood of the zero solution of a system of ordinary differential equations, Differential Equations, pp. 967–976, 1967.

  13. A. D. Bryuno, The normal form of a Hamiltonian system, Usp. Mat. Nauk, 43 (1988), 23–56, 247.

    Google Scholar 

  14. A. Chenciner, Bifurcations de points fixes elliptiques, Publ. Math., Inst. Hautes Étud. Sci., 61 (1985), 67–127.

    Google Scholar 

  15. E. M. Chirka, Complex analytic sets, vol. 46 of Mathematics and its Applications, Kluwer, 1989.

  16. L. H. Eliasson, Perturbations of stable invariant tori for Hamiltonian systems, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 15 (1988), 115–147.

  17. L. H. Eliasson, Absolutely convergent series expansions for quasi periodic motions, Math. Phys. Electron. J., 2, Paper 4, 33pp. (electronic), 1996.

  18. M. R. Herman, Sur les courbes invariantes par les difféomorphisme de l’anneau, vol. 1, Astérisque, 103–104 (1983), Société Mathématiques de France.

  19. M. R. Herman, Sur les courbes invariantes par les difféomorphisme de l’anneau, vol. 2, Astérisque, 144 (1986), Société Mathématiques de France.

  20. D. Y. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximations on manifolds, Ann. Math., 148 (1998), 339–360.

    Google Scholar 

  21. A. N. Kolmogorov, On the preservation of conditionally periodic motions under small variations of the hamilton function, Dokl. Akad. Nauk SSSR, 98 (1954), 527–530. English translation in “Selected Works”, Kluwer.

  22. A. N. Kolmogorov, The general theory of dynamical systems and classical mechanics, in Proceedings of International Congress of Mathematicians (Amsterdam, 1954), vol. 1, pp. 315–333, North-Holland, 1957, English translation in “Collected Works”, Kluwer.

  23. J. Moser, On invariant curves of aera-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II (1962), 1–20.

  24. J. Moser, Stable and random motions in dynamical systems, with special emphasis on celestian mechanics, vol. 77 of Ann. Math. Studies, Princeton University Press, 1973.

  25. H. Rüssmann, Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II (1970), 67–105.

  26. H. Rüssmann, Kleine Nenner II: Bemerkungen zur Newtonschen Methode, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II (1972), 1–10.

  27. H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001), 119–204.

    Google Scholar 

  28. C. L. Siegel and J. K. Moser, Lectures on Celestian Mechanics, Springer, 1971.

  29. S. Sternberg, Celestial Mechanics, Part I, W. A. Benjamin, 1969.

  30. S. Sternberg, Celestial Mechanics, Part II, W. A. Benjamin, 1969.

  31. L. Stolovitch, Complète intégrabilité singulière, C. R. Acad. Sci., Paris, Sér. I, Math., 326 (1998), 733–736.

  32. L. Stolovitch, Singular complete integrability, Publ. Math., Inst. Hautes Étud. Sci., 91 (2000), 133–210.

  33. L. Stolovitch, Un phénomène de type KAM, non symplectique, pour les champs de vecteurs holomorphes singuliers, C. R. Acad. Sci, Paris, Sér. I, Math., 332 (2001), 545–550.

  34. L. Stolovitch, Normalisation holomorphe d’algèbres de type Cartan de champs de vecteurs holomorphes singuliers, Ann. Math., 161 (2005), 589–612.

    Google Scholar 

  35. J.-C. Yoccoz, Birfurcations de points fixes elliptiques (d’après A. Chenciner), in Séminaire Bourbaki, Astérisque, 145–146 (1987), 313–334, Société Mathématiques de France, exposé 668.

  36. J.-C. Yoccoz, Travaux de Herman sur les tores invariants, in Séminaire Bourbaki, Astérisque, 206 (1992), 311–344, Société Mathématique de France, exposé 754.

  37. E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems I, Commun. Pure Appl. Math., 28 (1975), 91–140.

    Google Scholar 

  38. E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems II, Commun. Pure Appl. Math., 29 (1976), 49–111.

    Google Scholar 

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  1. Laboratoire de Mathématiques Emile Picard, Université Paul Sabatier, MIG, 31062, Toulouse cedex 9, France

    Laurent Stolovitch

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Stolovitch, L. A KAM phenomenon for singular holomorphic vector fields. Publ.math.IHES 102, 99–165 (2005). https://doi.org/10.1007/s10240-005-0035-0

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