Towards a semilocal study of parabolic invariant curves for fibred holomorphic maps

TL;DR

This paper investigates the local dynamics near parabolic invariant curves in fibred holomorphic maps, emphasizing the role of petals and base rotation number in the structure of the dynamics.

Contribution

It introduces a semilocal approach to study parabolic invariant curves in fibred holomorphic maps, highlighting the influence of base rotation number and providing a characterization theorem.

Findings

Existence and number of petals depend on base rotation number.

Examples show petals' properties are influenced by both map coordinate and rotation number.

A theorem characterizes local dynamics under certain arithmetic and smoothness conditions.

Abstract

We introduce the study of the local dynamics around a parabolic indifferent invariant curve for fibred holomorphic maps. As in the classical non-fibred case, we show that petals are the main ingredient. Nevertheless, one expects 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 hypothesis on the arithmetics and smoothness of the map, we present a theorem that allows to characterize the local dynamics around a parabolic invariant curve.