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We show that in the Gevrey topology, a $d$-torus flow close enough to linearwith a unique rotation vector $\omega$ is linearizable as long as $\omega$satisfies a Brjuno type diophantine condition. The proof is based on the fastconvergence under renormalization of the associated Gevrey vector field. Itrequires a multidimensional continued fractions expansion of $\omega$, and thecorresponding characterization of the Brjuno type vectors. This demonstratesthat renormalization methods deal very naturally with Gevrey regularityexpressed in the decay of Fourier coefficients. In particular, they providelinearization for frequencies beyond diophantine in non-analytic topologies.

Renormalization of Gevrey vector fields with a Brjuno type arithmetical condition