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Analytic invariants associated with a parabolic fixed point in ℂ2

Published online by Cambridge University Press:  15 October 2008

V. GELFREICH
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: V.Gelfreich@warwick.ac.uk)
V. NAUDOT
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK (email: V.Gelfreich@warwick.ac.uk)

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.

Type
Research Article
Copyright
Copyright © 2008 Cambridge University Press

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