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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The dynamics of maps tangent to the identity and with nonvanishing index


Author: Laura Molino
Journal: Trans. Amer. Math. Soc. 361 (2009), 1597-1623
MSC (2000): Primary 32H50, 37F10
DOI: https://doi.org/10.1090/S0002-9947-08-04533-9
Published electronically: October 22, 2008
MathSciNet review: 2457410
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Abstract: Let $ f$ be a germ of a holomorphic self-map of $ \mathbb{C}^2$ at the origin $ O$ tangent to the identity, and with $ O$ as a nondicritical isolated fixed point. A parabolic curve for $ f$ is a holomorphic $ f$-invariant curve, with $ O$ on the boundary, attracted by $ O$ under the action of $ f$. It has been shown by M. Abate (2001) that if the characteristic direction $ [v]\in\mathbb{P}(T_O\mathbb{C}^2)$ has residual index not belonging to $ \mathbb{Q}^+$, then there exist parabolic curves for $ f$ tangent to $ [v]$. In this paper we prove, using a different method, that the conclusion still holds just assuming that the residual index is not vanishing (at least when $ f$ is regular along $ [v]$).


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Additional Information

Laura Molino
Affiliation: Dipartimento di Matematica, Università degli Studi di Parma, Viale G. P. Usberti 53/A, I-43100, Parma, Italy
Email: laura.molino@unipr.it

DOI: https://doi.org/10.1090/S0002-9947-08-04533-9
Received by editor(s): April 8, 2005
Received by editor(s) in revised form: March 15, 2007
Published electronically: October 22, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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