Skip to main content
SpringerLink
Log in

Groups of Germs of Analytic Diffeomorphisms in (C2, 0)

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

Let G be a group of germs of analytic diffeomorphisms in (C2, 0). We find some remarkable properties supposing that G is finite, linearizable, abelian, nilpotent, and solvable. In particular, if the group is abelian and has a generic dicritic diffeomorphisms, then the group is a subgroup of a 1-parametric group. In addition, we study the topological behavior of the orbits of a dicritic diffeomorphisms. Last, we find some invariants in order to know when two diffeomorphisms are formally conjugate.

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. M. Abate, The residual index and the dynamics of holomorphic maps tangent to the identity. Duke Math. J. 107 (2001), No. 1, 173-206.

    Google Scholar 

  2. V. Arnold, Chapitres Suplémentaires de la théorie des équations différentielles ordinaires. Mir, Moscow, 1980.

    Google Scholar 

  3. C. Camacho. On the structure of conformal mappings and holomorphic vector fields in ℂ2. Astérisque 59–60 (1978), 83-94.

    Google Scholar 

  4. C. Camacho, A. Lins Neto, and P. Sad, Topological invariants and equidesingularization for holomorphic vector fields. J. Differential Geom. 20 (1984), 143-174.

    Google Scholar 

  5. J.C. Canille Martins, Holomorphic flows in (ℂ3, 0) with resonances. Trans. Amer. Math. Soc. 329 (1992), 825-837.

    Google Scholar 

  6. D. Cerveau and R. Moussu, Grupes d'automorphismes de (ℂ, 0) et equations differentielles y dy +...= 0. Bull. Soc. Math. France 116 (1988), 459-488.

    Google Scholar 

  7. J. Dixon, The structure of linear groups. Van Nostrand, 1971.

  8. J. écalle, Les fonctions résurgentes. Vol. I, II, and III. Preprint Mathématiques d'Orsay Université de Paris-Sud, France (1982–1985).

    Google Scholar 

  9. P.M. Elizarov, Yu. S. Il'yashenko, A.A. Shcherbakov, and S.M. Voronin, Finitely generated groups of germs of one-dimensional conformal mappings, and invariants for complex singular points of analytic foliations of the complex plane. Nonlinear Stokes phenomena. Adv. Sov. Math. 4 (1993), 57-105.

    Google Scholar 

  10. M. Hakim, Analytic transformations of (ℂp, 0) tangent to the identity. Duke Math. J. 92 (1998), 403-428.

    Google Scholar 

  11. M. Herman, Recent results and some open questions on Siegel's linearization theorem of germs of complex analytic diffeomorphics of ℂn near a fixed point. In: Proc. 8 Int. Conf. Math. Phys., World Scientific Publisher, Singapour (1987).

    Google Scholar 

  12. J. Martinet etJ.P. Ramis. Classification analytique des équations différentielles non linéares résonnantes du premier ordre. Ann. Sci. école Norm. Sup. 13 (1983), 571-621.

    Google Scholar 

  13. B. Malgrange, Introduction aux travaux de J. Ecalle. L'Enseignemet Mathématique 31 (1985), 261-282.

    Google Scholar 

  14. J.F. Mattei and R. Moussu, Holonomie et Intéegrales Premières. Ann. Sci. école Norm. Sup. 13 (1980).

  15. I. Nakai, Separatrices for nonsolvable dynamic on (ℂ, 0). In: Ann. Inst. Fourier, Genoble 44 (1994), No. 2, 569-599.

  16. M.F. Newman, The soluble lenght of soluble linear groups. Math. Z. 126 (1972), 59-70.

    Google Scholar 

  17. J.F. Plante, Lie algebras of vector fields which vinish at a point. J. London Math. Soc. 38 (1988), No. 2, 379-384.

    Google Scholar 

  18. R. Perez Marco, Solution complète au probleme de Siegel de linérisation d'une aplication holomorphe au voisinage d'un point fixe. Astérisque 206 (1992).

  19. T. Ueda, Local structure of analytic transformations of two complex variables, I. J. Math Kyoto Univ. 26 (1986), No. 2, 233-261.

    Google Scholar 

  20. T. Ueda, Local structure of analytic transformations of two complex variables, II. J. Math Kyoto Univ. 31 (1991), No. 3, 695-711.

    Google Scholar 

  21. S.M. Voronin, Analytic classification of germs of maps (ℂ, 0) → (ℂ, 0) with identical liner part. Funct. Anal. 15 (1981), 1-17.

    Google Scholar 

  22. S.M. Voronin, The Darboux-Whitney theorem and related questions. Non-linear Stokes Phenomena. Adv. Sov. Math. 14 (1993), 139-234.

    Google Scholar 

  23. W. Wasow, Asymptotic expansions for ordinary differential equation. Interscience Publ. (1965).

  24. B. A. F. Wehrfritz, Infinite linear groups. Springer-Verlag, 1973.

Download references

Author information

Authors and Affiliations

  1. Instituto de Matemática Pura e Aplicada, IMPA, Etrada Dona Castorina 110, CEP, 22460–320, Rio de Janeiro, Brasil

    Fabio Enrique Brochero Martínez

Authors
  1. Fabio Enrique Brochero Martínez
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martínez, F.E.B. Groups of Germs of Analytic Diffeomorphisms in (C2, 0). Journal of Dynamical and Control Systems 9, 1–32 (2003). https://doi.org/10.1023/A:1022101132569

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1022101132569

Search

Navigation