The dynamics of maps tangent to the identity and with nonvanishing index
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- by Laura Molino PDF
- Trans. Amer. Math. Soc. 361 (2009), 1597-1623 Request permission
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 $[v]$).References
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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
- Received by editor(s): April 8, 2005
- Received by editor(s) in revised form: March 15, 2007
- Published electronically: October 22, 2008
- © Copyright 2008 American Mathematical Society
- 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
- MathSciNet review: 2457410