Analytic invariants associated with a parabolic fixed point in C2
Gelfreich, Vassili and Naudot, V.. (2008) Analytic invariants associated with a parabolic fixed point in C2. Ergodic Theory and Dynamical Systems, Vol.28 (No.6). pp. 1815-1848. ISSN 0143-3857
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Official URL: http://dx.doi.org/10.1017/S0143385707001046
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.
|Item Type:||Journal Article|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Mathematics|
|Library of Congress Subject Headings (LCSH):||Parabolic operators, Diffeomorphisms, Invariants, Mappings (Mathematics), Differential topology|
|Journal or Publication Title:||Ergodic Theory and Dynamical Systems|
|Publisher:||Cambridge University Press|
|Official Date:||December 2008|
|Page Range:||pp. 1815-1848|
|Access rights to Published version:||Open Access|
|Funder:||Engineering and Physical Sciences Research Council (EPSRC)|
|Grant number:||EP/C000595/1 (EPSRC)|
 Algaba, A., Freire, E., Gamero, E. and García, C.. Quasi-homogeneous normal forms. J. Comput. Appl. Math. 150(1) (2003), 193–216.
 Gelfreich, V. G. and Sauzin, D.. Borel summation and the splitting of separatrices for the Hénon map. Ann. Inst. Fourier (Grenoble) 51(2) (2001), 513–567.
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