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arXiv · 1412.2615

Normal form of holomorphic vector fields with an invariant torus under Brjuno's A condition

Claire Chavaudret
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Abstract

We consider the holomorphic normalization problem for a holomorphic vector field in the neighborhood of the product of a fixed point and an invariant torus. Supposing that the vector field is a perturbation of a linear part around the fixed point and of a rotation on the invariant torus (the unperturbed vector field is called the quasi-linear part of the perturbed one), it was shown by J.Aurouet that the system is holomorphically linearizable if there are no exact resonances in the quasi-linear part and if the quasi-linear part satisfies to Brjuno's arithmetical condition. In the presence of exact resonances, a conjecture by Brjuno states that the system will still be holomorphically conjugated to a normal form under the same arithmetical condition and a strong algebraic condition on the formal normal form. This article proves this conjecture.

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Claire Chavaudret. 2016-02-10. Normal form of holomorphic vector fields with an invariant torus under Brjuno's A condition. https://arxiv.org/abs/1412.2615

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