Renormalisation of flows on the multidimensional torus close to a KT frequency vector

Abstract

We use a renormalisation operator R acting on a space of vector fields on t ᵈ , d ≥ 2, to prove the existence of a submanifold of vector fields equivalent to constant. The result comes from the existence of a fixed point ω of R which is hyperbolic. This is done for a certain class KTd of frequency vectors ω ∈ R ᵈ , called of Koch type. The transformation R is constructed using a time rescaling, a linear change of basis plus a periodic non-linear map isotopic to the identity, which we derive by a “homotopy trick”.