Let X be a germ of holomorphic vector field at the origin of 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},
volume = {102},
year = {2005},
pages = {99-165},
doi = {10.1007/s10240-005-0035-0},
mrnumber = {2217052},
zbl = {1114.37026},
language = {en},
url = {http://dml.mathdoc.fr/item/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://gdmltest.u-ga.fr/item/PMIHES_2005__102__99_0/
1. , 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 0135.42601
2. , 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 163025 | Zbl 0129.16606
3. , Small denominators and the problem of stability of motion in the classical and celestian mechanics, Russ. Math. Surv., 18 (1963), 85-191. | MR 170705 | Zbl 0135.42701
4. , Méthodes mathématiques de la mécanique classique, Mir, 1976. | MR 474391 | Zbl 0385.70001
5. , Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Mir, 1980. | MR 626685 | Zbl 0455.34001
6. (ed.), Dynamical systems III, vol. 28 of Encyclopaedia of Mathematical Sciences, Springer, 1988. | MR 923953 | Zbl 0623.00023
7. , A strengthened extremal property of Chebyshev polynomials, Moscow Univ. Math. Bull., 42 (1987), 24-26. | MR 884508 | Zbl 0645.33014
8. and , Metric diophantine approximation on manifolds, vol. 137 of Cambridge Tracts in Mathematics, Cambridge University Press, 1999. | MR 1727177 | Zbl 0933.11040
9. , , and , Quasi-periodic motions in famillies of dynamical systems, Lect. Notes Math. 1645, Springer, 1996. | MR 1484969 | Zbl 0870.58087
10. , Local theory of nonlinear analytic ordinary differential equations, Lect. Notes Math. 702, Springer, 1979. | MR 547669 | Zbl 0404.34005
11. , 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 0602.58021
12. and , 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 0233.34051
13. , The normal form of a Hamiltonian system, Usp. Mat. Nauk, 43 (1988), 23-56, 247. | MR 937018 | Zbl 0642.70009
14. , Bifurcations de points fixes elliptiques, Publ. Math., Inst. Hautes Étud. Sci., 61 (1985), 67-127. | Numdam | MR 783349 | Zbl 0566.58025
15. , Complex analytic sets, vol. 46 of Mathematics and its Applications, Kluwer, 1989. | MR 1111477 | Zbl 0683.32002
16. , Perturbations of stable invariant tori for Hamiltonian systems, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 15 (1988), 115-147. | Numdam | MR 1001032 | Zbl 0685.58024
17. , Absolutely convergent series expansions for quasi periodic motions, Math. Phys. Electron. J., 2, Paper 4, 33pp. (electronic), 1996. | MR 1399458 | Zbl 0896.34035
18. , Sur les courbes invariantes par les difféomorphisme de l'anneau, vol. 1, Astérisque, 103-104 (1983), Société Mathématiques de France. | Zbl 0532.58011
19. , Sur les courbes invariantes par les difféomorphisme de l'anneau, vol. 2, Astérisque, 144 (1986), Société Mathématiques de France. | Zbl 0613.58021
20. and , Flows on homogeneous spaces and Diophantine approximations on manifolds, Ann. Math., 148 (1998), 339-360. | MR 1652916 | Zbl 0922.11061
21. , 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 0056.31502
22. , 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. , On invariant curves of aera-preserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II (1962), 1-20. | MR 147741 | Zbl 0107.29301
24. , Stable and random motions in dynamical systems, with special emphasis on celestian mechanics, vol. 77 of Ann. Math. Studies, Princeton University Press, 1973. | MR 442980 | Zbl 0271.70009
25. , Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II (1970), 67-105. | MR 273156 | Zbl 0201.11202
26. , Kleine Nenner II: Bemerkungen zur Newtonschen Methode, Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II (1972), 1-10. | MR 309297 | Zbl 0255.30003
27. , Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001), 119-204. | MR 1843664 | Zbl 0992.37050
28. and , Lectures on Celestian Mechanics, Springer, 1971. | MR 502448 | Zbl 0817.70001
29. , Celestial Mechanics, Part I, W. A. Benjamin, 1969. | Zbl 0293.70013
30. , Celestial Mechanics, Part II, W. A. Benjamin, 1969. | Zbl 0194.56702
31. , Complète intégrabilité singulière, C. R. Acad. Sci., Paris, Sér. I, Math., 326 (1998), 733-736. | MR 1641778 | Zbl 0917.32029
32. , Singular complete integrability, Publ. Math., Inst. Hautes Étud. Sci., 91 (2000), 133-210. | Numdam | MR 1828744 | Zbl 0997.32024
33. , 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 1834067 | Zbl 0997.32025
34. , Normalisation holomorphe d'algèbres de type Cartan de champs de vecteurs holomorphes singuliers, Ann. Math., 161 (2005), 589-612. | Zbl 1080.32019
35. , 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 0616.58035
36. , Travaux de Herman sur les tores invariants, in Séminaire Bourbaki, Astérisque, 206 (1992), 311-344, Société Mathématique de France, exposé 754. | Numdam | MR 1206072 | Zbl 0791.58044
37. , Generalized implicit function theorems with applications to some small divisor problems I, Commun. Pure Appl. Math., 28 (1975), 91-140. | MR 380867 | Zbl 0309.58006
38. , Generalized implicit function theorems with applications to some small divisor problems II, Commun. Pure Appl. Math., 29 (1976), 49-111. | MR 426055 | Zbl 0334.58009