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Towards a semi-local study of parabolic invariant curves for fibered holomorphic maps

Published online by Cambridge University Press:  14 October 2011

MARIO PONCE
MARIO PONCE*
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Macul, Santiago, Chile (email: mponcea@mat.puc.cl)
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Abstract

We introduce the study of the local dynamics around a parabolic indifferent invariant curve for fibered holomorphic maps. As in the classical non-fibered case, we show that petals are the main ingredient. Nevertheless, one expects that the properties of the base rotation number should play an important role in the arrangement of the petals. We exhibit examples where the existence and the number of petals depend not just on the complex coordinate of the map, but on the base rotation number. Furthermore, under additional hypotheses on the arithmetic and smoothness of the map, we present a theorem that allows a characterization of the local dynamics around a parabolic invariant curve.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 6 , December 2012 , pp. 2056 - 2070
Copyright
Copyright © Cambridge University Press 2011

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