zbMATH — the first resource for mathematics

Singular complete integrability. (English) Zbl 0997.32024
This very interesting paper is concerned with the study of holomorphic vector fields in a neighbourhood of a singular point in \(\mathbb C^{n}\). Such a vector field is holomorphically normalizable if its linear part is not too wild and has enough formal first integrals as symmetries.
In order to prove this kind of result the author studies collections of vector fields which commute pairwise and are described by a Lie morphism from a complex commutative finite-dimensional Lie algebra to the algebra of vector fields in a neighbourhood of the singular point which is the origin of \(\mathbb C^{n}\). The result explains and generalizes some previous results e.g. of A. D. Bryuno [Trans. Mosc. Math. Soc. 25, 131-288 (1973); translation from Tr. Mosk. Mat. Obshch. 25, 119-226 (1971; Zbl 0263.34003) and ibid. 26, 199-239 (1974); translation from Tr. Mosk. Mat. Obshch. 26, 199-239 (1972; Zbl 0269.34003)] and J. Vey [Bull. Soc. Math. Fr. 107, 423-432 (1979; Zbl 0426.58022)]. An interesting geometric interpretation of the results is also given.

32S65 Singularities of holomorphic vector fields and foliations
37F75 Dynamical aspects of holomorphic foliations and vector fields
17B66 Lie algebras of vector fields and related (super) algebras
17B80 Applications of Lie algebras and superalgebras to integrable systems
32G99 Deformations of analytic structures
Full Text: DOI Numdam EuDML
[1] V. Arnold,Méthodes mathématiques de la mécanique classique. Mir (1976).
[2] V. Arnold,Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Mir (1980).
[3] N. Bourbaki,Groupes et Algèbres de Lie, Chapitres 7 et 8. Paris, Masson (1990).
[4] M. Brion, Invariants et coinvariants des groupes algébriques réductifs. Notes d’un cours à l’école d’été de Monastir, juillet–août 1996.
[5] A. D. Bruno, The analytical form of differential equations.Trans. Mosc. Math. Soc. 25 (1971), 131–288;26 (1972), 199–239.
[6] A. D. Bruno andS. Walcher, Symmetries and convergence of normalizing transformations.J. Math. Analysis Appl.,183 (1994), 571–576. · Zbl 0804.34040
[7] H. Cartan andS. Eilenberg,Homological algebra. Princeton University Press (1956). · Zbl 0075.24305
[8] D. Cerveau, Distributions involutives singulières.Ann. Inst. Fourier, Grenoble,29(3) (1979), 261–294. · Zbl 0419.58002
[9] G. Cairns andE. Ghys, The local linearlization problem for smoothsl(n)-actions.Enseignement Math.,43 (1997). · Zbl 0914.57027
[10] M. Chaperon, Géométrie différentielle et singularités de systèmes dynamiques.Astérisque,138–139 (1986).
[11] C. Camacho andP. Sad,Pontos singulares de equações diferenciais analíticas. 16 Colóquio Brasileiro de Matemática.
[12] D. Delatte, Diophantine conditions for the linearization of commuting holomorphic functions.Discrete and Continuous Dyn. Sys.,3(3) (1997), 317–332. · Zbl 0949.30020
[13] F. Dumortier andR. Roussarie, Smooth linearization of germs ofR 2-actions and holomorphic vector fields.Ann. Inst. Fourier, Grenoble,30(1) (1980), 31–64. · Zbl 0418.58015
[14] J. Ecalle, Sur les fonctions résurgentes. I, II, III Publ. Math. d’Orsay.
[15] J. Ecalle, Singularités non abordables par la géométrie.Ann. Inst. Fourier, Grenoble,42, 1–2 (1992), 73–164.
[16] J. P. Françoise, Singularités de champs isochores.Duke Math. Journ.,67,3 (1980), p. 665–685.
[17] H. Grauert andR. Remmert,Analytische Stellenalgebren (1971), Springer-Verlag.
[18] V. V. Guillemin andS. Sternberg, Remarks on a paper of Hermann.Trans. Amer. Math. Soc.,130 (1968), 110–116. · Zbl 0155.05701
[19] T. Gramchev andM. Yoshino, Rapidly convergent iteration method for simultaneous normal forms of commuting maps.Math. Z.,231 (1999), 745–770. · Zbl 0931.65055
[20] R. Hermann, The formal linearization of a semi-simple Lie algebra of vector fields about a singular point.Trans. Amer. Math. Soc.,130 (1968), 105–109. · Zbl 0155.05604
[21] H. Ito, Convergence of Birkhoff normal forms for integrable systems.Comment. Math. Helv.,64 (1989), 412–461. · Zbl 0686.58021
[22] H. Ito, Integrability of hamiltonian systems and Birkhoff normal forms in the simple resonance case.Math. Ann.,292 (1992), 411–444. · Zbl 0735.58022
[23] A. G. Kushnirenko, Linear-equivalent action of a semi-simple Lie group in the neighbourhood of a stationary point.Funct. Anal. Appl.,1 (1967), 89–90. · Zbl 0156.42205
[24] V. V. Lychagin, Singularities of solutions, spectral sequences and normal forms of Lie algebras of vector fields.Math. USSR Izvestiya,30(3) (1988), 549–575. · Zbl 0675.58044
[25] B. Malgrange, Frobenius avec singularités, 1. Codimension 1.Publ. Math. I.H.E.S,46 (1976), 163–173. · Zbl 0355.32013
[26] B. Malgrange, Frobenius avec singularités, 2. Cas général.Invent. Math.,39 (1977), 67–89. · Zbl 0375.32012
[27] B. Malgrange, Travaux d’Ecalle et de Martinet-Ramis sur les systèmes dynamiques.Séminaire Bourbaki 1981–1982, exp. 582,Astérisque 92–93 (1982).
[28] J. Martinet, Normalisation des champs de vecteurs holomorphes.Séminaire Bourbaki 1980–1981, exposé 564, 901,Lecture Notes in Mathematics,55–70 (1981), Springer-Verlag.
[29] J.-F. Mattei andR. Moussu, Intégrales premières et holonomie.Ann. scient. Éc. Norm. Sup.,13 (1980), 469–523.
[30] J. Moser, On commuting circle mappings and simultaneous diophantine approximations.Math. Z.,205 (1990), 105–121. · Zbl 0689.58031
[31] J. Martinet andJ. P. Ramis, Problèmes de modules pour des équations différentielles non linéaires du premier ordre.Publ. Math. I.H.E.S,55 (1982), 63–164. · Zbl 0546.58038
[32] J. Martinet andJ. P. Ramis, Classification analytique des équations différentielles non linéaires résonnantes du premier ordre.Ann. scient. Éc. Norm. Sup., 4e série,16 (1983), 571–621.
[33] J. P. Ramis, Frobenius avec singularités d’après B. Malgrange, J.-F. Mattei et R. Moussu.Séminaire Bourbaki, 1977/1978, exposé 523, 710,Lecture Notes in Mathematics,290–299 (1979), Springer-Verlag.
[34] R. Roussarie, Modèles locaux de champs et de formes.Astérisque,30 (1975).
[35] J.-P. Serre,Lie Algebras and Lie groups, 1500,Lecture Notes in Mathematics, (1992), Springer-Verlag.
[36] C. L. Siegel, Iterations of analytic functions.Ann. Math.,43 (1942), 807–812.
[37] L. Stolovitch, Classification analytique de champs de vecteurs 1-résonnants de (C n , 0).Asymptotic Analysis,12 (1996), 91–143. · Zbl 0852.58013
[38] L. Stolovitch, Forme normale de champs de vecteurs commutants.C.R. Acad. Sci, Paris, Série I,324 (1997) 665–668. · Zbl 0885.58081
[39] L. Stolovitch, Complète intégrabilité singulière.C.R. Acad. Sci., Paris, Série I,326 (1998), 733–736. · Zbl 0917.32029
[40] L. Stolovitch, Singular complete integrability. Technical Report 111, Prépublication E. Picard, janvier 1998, 1–37. · Zbl 0917.32029
[41] L. Stolovitch, Normalisation holomorphe d’algèbres de type Cartan de champs de vecteurs holomorphes singuliers.C.R. Acad. Sci., Paris, Série I,330 (2000), 121–124. · Zbl 1039.32042
[42] L. Stolovitch, Normalisation holomorphe d’algèbres de type Cartan de champs de vecteurs holomorphes singuliers.Technical Report 186, Prépublication E. Picard, mars 2000, 1–27. · Zbl 1039.32042
[43] J. C. Tougeron,Idéaux de fonctions différentiables. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71 (1972), Springer-Verlag.
[44] J. Vey, Sur certains systèmes dynamiques séparables.Am. Journal of Math.,100 (1978), p. 591–614. · Zbl 0384.58012
[45] J. Vey, Algèbres commutatives de champs de vecteurs isochores.Bull. Soc. Math. France,107 (1979), p. 423–432. · Zbl 0426.58022
[46] S. M. Voronin, Analytic classification of germs of conformal mappings (C, 0) (C, 0) with identity linear part.Funct. An. and its Appl.,15 (1981). · Zbl 0463.30010
[47] S. Walcher, On differential equations in normal form.Math. Ann.,291 (1991), 293–314. · Zbl 0754.34032
[48] J.-C. Yoccoz, Linéarisation des germes de difféomorphismes holomorphes de (C, 0).C.R. Acad. Sci. Paris, Série I,306 (1988), 55–58. · Zbl 0668.58010
[49] J.-C. Yoccoz, Petits diviseurs en dimension 1.Astérisque,231 (1995).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.