A KAM phenomenon for singular holomorphic vector fields
Publications Mathématiques de l'IHÉS, Tome 102 (2005), pp. 99-165.

Let X be a germ of holomorphic vector field at the origin of 𝐂 n 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.

@article{PMIHES_2005__102__99_0,
     author = {Stolovitch, Laurent},
     title = {A {KAM} phenomenon for singular holomorphic vector fields},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {99--165},
     publisher = {Springer},
     volume = {102},
     year = {2005},
     doi = {10.1007/s10240-005-0035-0},
     mrnumber = {2217052},
     zbl = {1114.37026},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1007/s10240-005-0035-0/}
}
TY  - JOUR
AU  - Stolovitch, Laurent
TI  - A KAM phenomenon for singular holomorphic vector fields
JO  - Publications Mathématiques de l'IHÉS
PY  - 2005
SP  - 99
EP  - 165
VL  - 102
PB  - Springer
UR  - http://archive.numdam.org/articles/10.1007/s10240-005-0035-0/
DO  - 10.1007/s10240-005-0035-0
LA  - en
ID  - PMIHES_2005__102__99_0
ER  - 
%0 Journal Article
%A Stolovitch, Laurent
%T A KAM phenomenon for singular holomorphic vector fields
%J Publications Mathématiques de l'IHÉS
%D 2005
%P 99-165
%V 102
%I Springer
%U http://archive.numdam.org/articles/10.1007/s10240-005-0035-0/
%R 10.1007/s10240-005-0035-0
%G en
%F PMIHES_2005__102__99_0
Stolovitch, Laurent. A KAM phenomenon for singular holomorphic vector fields. Publications Mathématiques de l'IHÉS, Tome 102 (2005), pp. 99-165. doi : 10.1007/s10240-005-0035-0. http://archive.numdam.org/articles/10.1007/s10240-005-0035-0/

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. | Zbl

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. | MR | Zbl

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. | MR | Zbl

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

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

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

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

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

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. | MR | Zbl

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

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. | Numdam | Zbl

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. | Zbl

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

14. A. Chenciner, Bifurcations de points fixes elliptiques, Publ. Math., Inst. Hautes Étud. Sci., 61 (1985), 67-127. | Numdam | MR | Zbl

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

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. | Numdam | MR | Zbl

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

18. M. R. Herman, Sur les courbes invariantes par les difféomorphisme de l'anneau, vol. 1, Astérisque, 103-104 (1983), Société Mathématiques de France. | Numdam | Zbl

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

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

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. | Zbl

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. | MR | Zbl

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. | MR | Zbl

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

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

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

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

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

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

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

32. L. Stolovitch, Singular complete integrability, Publ. Math., Inst. Hautes Étud. Sci., 91 (2000), 133-210. | Numdam | MR | Zbl

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. | MR | Zbl

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

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. | Numdam | Zbl

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. | Numdam | MR | Zbl

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

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

Cité par Sources :