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On the Persistence of Invariant Curves for Fibered Holomorphic Transformations

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

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  1. Pontificia Universidad Católica de Chile, Santiago, Chile

    Mario Ponce

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  1. Mario Ponce
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Correspondence to Mario Ponce.

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Communicated by G. Gallavotti

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Ponce, M. On the Persistence of Invariant Curves for Fibered Holomorphic Transformations. Commun. Math. Phys. 289, 1–44 (2009). https://doi.org/10.1007/s00220-009-0805-5

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