Brjuno condition through best approximations and the linearization problem

2026/07/28 by Nicolas Chevallier, João Lopes Dias, José Pedro Gaivão +2
#math.DS

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Abstract

We consider the classical analytic linearization problem for vector fields on the torus \mathbbT^d close to a constant vector field ω. Our goals are twofold. First, we provide a geometric framework in which the arithmetic condition governing analytic linearization arises naturally from the orbit of a unimodular lattice associated with ω under a diagonal flow on SL(d,ℤ)\backslash SL(d,ℝ). Within this framework, a summability condition emerges as the natural criterion for convergence. We prove that it is equivalent to several classical formulations of the Brjuno condition for linear forms, including those involving best approximation vectors and switching times of the diagonal flow. As a byproduct, we obtain a new quantitative linearization theorem with fully explicit estimates. In particular, the loss of analyticity of the conjugacy is controlled by a Brjuno function.

Citations

• On the linearization of analytic diffeomorphisms of the torus
• Optimal linearization of vector fields on the torus in non-analytic Gevrey classes
• Local conjugacy classes for analytic torus flows
• KAM Theorem for Gevrey Hamiltonians
• Rotation Sets for Maps of Tori

Related

• Renormalization of Gevrey vector fields with a Brjuno type arithmetical condition
• Optimal linearization of vector fields on the torus in non-analytic Gevrey classes
• Local conjugacy classes for analytic torus flows
• Analytic linearization of a generalization of the semi-standard map: radius of convergence and Brjuno sum
• On the linearization of analytic diffeomorphisms of the torus