Skip to main content
SpringerLink
Log in

Dynamics and Integrability for Germs of Complex Vector Fields with Singularity in Dimension Two

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper we prove classification results for singularities of holomorphic vector fields under some dynamical hypotheses on their orbits regarding their separatrices. We consider germs for which the set of separatrices is finite. We prove that if there is no orbit accumulating properly at the set of separatrices then the corresponding holomorphic foliation admits a holomorphic first integral or it is a linear logarithmic foliation of real type (cf. Theorem 1.1). Then we conclude (cf. Theorem 1.1 and Corollary 1.2) that the germ of a holomorphic foliation, with a finite number of separatrices, admits a holomorphic first integral iff it admits some closed leaf and no other leaf accumulates at the singularity. More generally, we prove (cf. Theorem 1.5) that if the germ admits some closed leaf then either it admits a holomorphic first integral or it is associated to a Pérez-Marco singularity (see Example 1.4). In the second case the germ has infinitely many closed leaves, the set of leaves accumulating properly at the set of separatrices (and therefore at the singularity) is not empty, and the singularity is essentially formally linearizable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexander, J.C., Verjovsky, A.: First integrals for singular holomorphic foliations with leaves of bounded volume, Holomorphic dynamics (Mexico, 1986), 1–10, Lecture Notes in Math. 1345, Springer-Verlag, Berlin (1988)

  2. Biswas, Kingshook.: Simultaneous linearization of germs of commuting holomorphic diffeomorphisms. Ergod. Theory Dyn. Syst. 32, 1216–1225 (2012). doi:10.1017/S0143385711000307

  3. Bracci, F.: Local dynamics of holomorphic diffeomorphisms. Boll. UMI 7-B , pp 609–636 (2004)

  4. Camacho, C.: On the local structure of conformal maps and holomorphic vector fields in \({\mathbb{C}}^{2}\), Jounées Singuliéres de Dijon (Univ. Dijon, Dijon, 1978, 3, 83–94). Astérisque pp 59–60, Soc. Math. France, Paris (1978)

  5. Camacho, C., Kuiper, N., Palis, J.: The topology of holomorphic flows with singularity. Inst. Hautes Études Sci. Publ. Math. 48, 5–38 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Camacho, C., Lins Neto, A.: Geometric theory of foliations, Birkhauser Inc., Boston, Mass (1985)

  7. Camacho, C., Lins Neto, A., Sad, P.: Foliations with algebraic limit sets. Ann. Math. 136(2), 429–446 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Camacho, C., Rosas, R.: Invariant sets near singularities of holomorphic foliations, arXiv:1312.0927 [math.DS] (or arXiv:1312.0927v1 [math.DS] for the newest version).

  9. Camacho, C., Sad, P.: Invariant varieties through singularities of holomorphic vector fields. Ann. Math. 115, 579–595 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Camacho, C., Sad, P.: Topological classification and bifurcations of holomorphic flows with resonances in \({\mathbb{C}}^{2}\). Invent. Math. 67, 447–472 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  11. Camacho, C., Scárdua, B.: Foliations on complex projective spaces with algebraic limit sets. In: Géométrie complexe et systèmes dynamiques (Orsay, 1995). Astérisque, 261, pp 57–88. Soc. Math, Paris (2000)

  12. Camacho, C., Scárdua, B.: On the existence of compact leaves for transversely holomorphic foliations. Topol. Appl. 160(13), 1802–1808 (2014)

  13. Camacho, C., Scárdua, B.: A Darboux type theorem for germs of complex vector fields. Pre-print IMPA/UFRJ (2013)

  14. Cerveau, D., Moussu, R.: Groupes d’automorphismes de \(({\mathbb{C}},0)\) et équations différentielles ydy + \(\cdots \) = 0. Bull. Soc. Math. France 116, 459–488 (1988)

    MathSciNet  MATH  Google Scholar 

  15. Cremer, H.: Zum Zentrumproblem. Math. Ann. 98, 151–163 (1928)

  16. Dulac, H.: Solutions d’un système de équations différentiale dans le voisinage des valeus singulières. Bull. Soc. Math. France 40, 324–383 (1912)

    MathSciNet  MATH  Google Scholar 

  17. Gunning, R.C.: Introduction to holomorphic functions of several variables. Function theory, vol. I. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove (1990)

    Google Scholar 

  18. Gunning, R.C.: Introduction to holomorphic functions of several variables. Local theory, vol. II. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey (1990)

    Google Scholar 

  19. Jouanolou, J.P.: Équations de Pfaff algèbriques, Lecture Notes in Math. 708. Springer, Berlin (1979)

    Google Scholar 

  20. Marín, D., Mattei, J.-F.: Mapping class group of a plane curve germ. Topol. Appl. 158, 1271–1295 (2011)

    Article  MATH  Google Scholar 

  21. Mattei, J.F., Moussu, R.: Holonomie et intégrales premières. Ann. Sci. École Norm. Sup. 13(4), 469–523 (1980)

    MathSciNet  MATH  Google Scholar 

  22. Martinet, J., Ramis, J.-P.: Classification analytique des équations différentielles non linéaires resonnants du premier ordre. Ann. Sci. École Norm. Sup. 16(4), 571–621 (1983)

    MathSciNet  MATH  Google Scholar 

  23. Martinet, J., Ramis, J.-P.: Problème de modules pour des équations différentielles non linéaires du premier ordre. Publ. Math. Inst. Hautes Études Scientifiques 55, 63–124 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nakai, I.: Separatrices for nonsolvable dynamics on \({\mathbb{C}},0\). Ann. Inst. Fourier (Grenoble) 44, 569–599 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pérez-Marco, R.: Sur les dynamiques holomorphes non lin’earisables et une conjecture de V.I. Arnold. Ann. Sci. École Norm. Sup. 26, 644 (1993)

    Google Scholar 

  26. Pérez-Marco, R.: Sur une question de Dulac et Fatou. C. R. Acad. Sci. Paris Ser. I Math. 321, 1045–1048 (1995)

  27. Pérez-Marco, R.: Fixed points and circle maps. Acta Math. 179, 243–294 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Jasmin Raissy.: Geometrical methods in the normalization of germs of biholomorphisms, Ph.D. Thesis, Università di Pisa 2010 (available at http://www.math.univ-toulouse.fr/jraissy/papers/papers.html)

  29. Scárdua, B.: Complex vector fields having orbits with bounded geometry, Tohoku Math. J. (2) 54(3), 367–392 (2002)

  30. Scárdua, B.: Integration of complex differential equations. J. Dyn. Control Syst. 1(5), 1–50 (1999)

    Article  Google Scholar 

  31. Seidenberg, A.: Reduction of singularities of the differential equation Ady = Bdx. Am. J. Math. 90, 248–269 (1968)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

I am very grateful to the referee for the careful reading and valuable suggestions, which helped improving the paper.

Author information

Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal do Rio de Janeiro, CP. 68530, Rio de Janeiro, RJ, 21945-970, Brazil

    Bruno Scárdua

Authors
  1. Bruno Scárdua
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bruno Scárdua.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Scárdua, B. Dynamics and Integrability for Germs of Complex Vector Fields with Singularity in Dimension Two. Qual. Theory Dyn. Syst. 13, 363–381 (2014). https://doi.org/10.1007/s12346-014-0122-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-014-0122-z

Mathematics Subject Classification

Search

Navigation