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Characteristic directions of two-dimensional biholomorphisms

Part of: Holomorphic mappings and correspondences Singularities Complex dynamical systems

Published online by Cambridge University Press:  31 March 2020

Lorena López-Hernanz  and
Rudy Rosas
Lorena López-Hernanz
Affiliation:
Departamento de Física y Matemáticas, Universidad de Alcalá, Edificio de Ciencias, Carretera Madrid-Barcelona, Km. 33600, 28871Alcalá de Henares, Madrid, Spain email lorena.lopezh@uah.es
Rudy Rosas
Affiliation:
Departamento de Ciencias, Pontificia Universidad Católica del Perú, Av. Universitaria 1801, Lima, Peru email rudy.rosas@pucp.pe
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Abstract

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $(\mathbb{C}^{2},0)$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.

Keywords

local holomorphic dynamicstangent to the identity mapscharacteristic directionsparabolic manifolds

MSC classification

Primary: 32H50: Iteration problems 37F99: None of the above, but in this section 32S65: Singularities of holomorphic vector fields and foliations
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 5 , May 2020 , pp. 869 - 880
Copyright
© The Authors 2020

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Footnotes

The first author was partially supported by Ministerio de Economía y Competitividad, Spain, process MTM2016-77642-C2-1-P; the second author was supported by Vicerrectorado de Investigación de la Pontificia Universidad Católica del Perú.

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