Normalization, optimal regularity, and solvability in Gevrey classes of vector fields near trapped orbits

Abstract

We prove a normalization and solvability result in Gevrey spaces for a family of vector fields in a neighborhood of a torus, extending recent work of Meziani [Trans. Amer. Math. Soc. 369 (2017), pp. 3325–3354]. The consideration of Gevrey order allows for a precise characterization of solvability, or lack thereof, for some vector fields of this class, which includes the real-analytic classes considered by Meziani. We also prove semiglobal solvability for a normalized family of real vector fields in the case of classical and non-homogeneous Gevrey spaces.