Renormalization of Gevrey vector fields with a Brjuno type arithmetical condition

TL;DR

This paper proves that certain Gevrey regular torus flows with frequencies satisfying a Brjuno type condition can be linearized, extending classical results beyond Diophantine frequencies using renormalization techniques.

Contribution

It introduces a renormalization approach to linearize Gevrey vector fields on tori under a Brjuno type condition, broadening the scope beyond Diophantine frequencies.

Findings

Linearization holds for Gevrey vector fields with Brjuno type frequencies.

Renormalization converges rapidly in the Gevrey topology.

Method extends linearization results beyond Diophantine frequencies.

Abstract

We show that in the Gevrey topology, a $d$-torus flow close enough to linear with a unique rotation vector $\omega$ is linearizable as long as $\omega$ satisfies a Brjuno type diophantine condition. The proof is based on the fast convergence under renormalization of the associated Gevrey vector field. It requires a multidimensional continued fractions expansion of $\omega$, and the corresponding characterization of the Brjuno type vectors. This demonstrates that renormalization methods deal very naturally with Gevrey regularity expressed in the decay of Fourier coefficients. In particular, they provide linearization for frequencies beyond diophantine in non-analytic topologies.