Location of Repository

Analytic invariants associated with a parabolic fixed point in C2

By Vassili Gelfreich and V. Naudot

Abstract

It is well known that in a small neighbourhood of a parabolic fixed point a real-analytic diffeomorphism of (R2,0) embeds in a smooth autonomous flow. In this paper we show that the complex-analytic situation is completely different and a generic diffeomorphism cannot be embedded in an analytic flow in a neighbourhood of its parabolic fixed point. We study two analytic invariants with respect to local analytic changes of coordinates. One of the invariants was introduced earlier by one of the authors. These invariants vanish for time-one maps of analytic flows. We show that one of the invariants does not vanish on an open dense subset. A complete analytic classification of the maps with a parabolic fixed point in C2 is not available at the present time

Topics: QA
Publisher: Cambridge University Press
Year: 2008
OAI identifier: oai:wrap.warwick.ac.uk:860

Suggested articles

Preview

Citations

  1. (1999). A proof of the exponentially small transversality of the separatrices for the standard map. doi
  2. (1992). An analytic integral along the separatrix of the semistandard map: existence and an exponential estimate for the distance between the stable and unstable separatrices.
  3. (1981). Analytic classification of germs of conformal mappings .C; 0/ ! doi
  4. (2001). Borel summation and the splitting of separatrices for the H´ enon map. doi
  5. (2001). Chaotic zone in the Bogdanov–Takens bifurcation for diffeomorphisms. Analysis and Applications, doi
  6. (1996). Conjugation to a shift and splitting of separatrices. doi
  7. (1973). Forced oscillations and bifurcations. Applications of Global Analysis. I (Symp., doi
  8. (1992). Further reduction of the Takens–Bogdanov normal form. doi
  9. (1996). Invariant circles in the Bogdanov–Takens bifurcation for diffeomorphisms. doi
  10. (1973). Nature du groupe des ordres d’it´ eration complexes d’une transformation holomorphe au voisinage d’un point fixe de multiplicateur 1.
  11. (2003). Quasi-homogeneous normal forms. doi
  12. (2000). Splitting of a small separatrix loop near the saddle-center bifurcation in area-preserving maps. doi
  13. (2002). Unique normal forms for nilpotent planar vector fields. doi
  14. (1975). Versal deformations of a singular point on the plane in the case of zero eigenvalues. doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.