Linearization of Gevrey flows on Td with a Brjuno type arithmetical condition

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 novel 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 new linearization results including frequencies beyond diophantine in non-analytic topologies. (C) 2019 Elsevier Inc. All rights reserved.