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

TL;DR

This paper proves Brjuno's conjecture that holomorphic vector fields with an invariant torus can be normalized under Brjuno's arithmetical condition even with exact resonances, extending previous linearization results.

Contribution

It establishes the validity of Brjuno's conjecture for holomorphic vector fields with invariant tori in the presence of exact resonances.

Findings

Proves Brjuno's conjecture on normal form under arithmetical conditions.

Extends linearization results to cases with exact resonances.

Confirms the applicability of Brjuno's condition in broader contexts.

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.