2014/12/08 by Claire Chavaudret, Chavaudret, Claire
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1412.2615
openalex publication_date 2014/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the holomorphic normalization problem for a holomorphic vector\nfield in the neighborhood of the product of a fixed point and an invariant\ntorus. Supposing that the vector field is a perturbation of a linear part\naround the fixed point and of a rotation on the invariant torus (the\nunperturbed vector field is called the quasi-linear part of the perturbed one),\nit was shown by J.Aurouet that the system is holomorphically linearizable if\nthere are no exact resonances in the quasi-linear part and if the quasi-linear\npart satisfies to Brjuno's arithmetical condition. In the presence of exact\nresonances, a conjecture by Brjuno states that the system will still be\nholomorphically conjugated to a normal form under the same arithmetical\ncondition and a strong algebraic condition on the formal normal form. This\narticle proves this conjecture.\n