On the persistence of invariant curves for Fibered Holomorphic Transformations

TL;DR

This paper investigates the conditions under which invariant curves in fibered holomorphic transformations persist when the system is slightly perturbed, focusing on the role of a fibered rotation number and a Brjuno-type arithmetic condition.

Contribution

It establishes a persistence result for invariant curves with prescribed fibered rotation numbers under small perturbations, extending the understanding of invariant structures in fibered holomorphic systems.

Findings

Invariant curves persist under small perturbations if the fibered rotation number satisfies a Brjuno-type condition.

The paper defines a fibered rotation number associated with invariant curves.

Persistence is proven for systems meeting specific arithmetical conditions on rotation numbers.

Abstract

We consider the problem of the persistence of invariant curves for analytical fibered holomorphic transformations. We define a fibered rotation number associated to an invariant curve. We show that an invariant curve with a prescribed fibered rotation number persists under small perturbations on the dynamics provided that the pair of rotation numbers verifies a Brjuno type arithmetical condition.