Renormalization of vector fields

Abstract

Abstract. These notes cover some of the recent developments in the renormalization of quasiperiodic flows. This includes skew flows over tori, Hamiltonian flows, and other flows on T d ×R . After stating some of the problems and describing alternative approaches, we focus on the definition and basic properties of a single renormalization step. A second part deals with the construction of conjugacies and invariant tori, including shearless tori, and non-differentiable tori for critical Hamiltonians. Then we discuss properties related to the spectrum of the linearized renormalization transformation, such as the accumulation rates for sequences of closed orbits. The last part describes extensions from "self-similar" to Diophantine rotation vectors. This involves sequences of renormalization transformations that are related to continued fractions expansions in one and more dimensions. Whenever appropriate, the discussion of details is restricted to special cases where inessential technical complications can be avoided.