On the Persistence of Invariant Curves for Fibered Holomorphic Transformations

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. Nevertheless, an extra complex parameter is added to the problem and the persistence becomes a one-complex codimension property.