Brjuno condition through best approximations and the linearization problem

Abstract

We consider the classical analytic linearization problem for vector fields on the torus $\mathbb{T}^d$ close to a constant vector field $\omega$. 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 $\omega$ under a diagonal flow on $\operatorname{SL}(d,\mathbb{Z})\backslash \operatorname{SL}(d,\mathbb{R})$. 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.