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

Abstract

AbstractWe 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.