A renormalization approach to lower-dimensional tori with Brjuno frequency vectors

Abstract

We extend the renormalization scheme for vector fields on T d ×R m in order to construct lower-dimensional invariant tori with Brjuno frequency vectors for near-integrable Hamiltonian flows. For every Brjuno frequency vector ω ∈ R d and every vector Ω ∈ R D satisfying a Diophantine condition with respect to ω, there exists an analytic manifold W of infinitely renormalizable Hamiltonian vector fields; each vector field on W is shown to have an analytic invariant torus with frequency vector ω.

Resonant motions in the presence of degeneracies for quasi-periodically perturbed systems

“…This introduces nontrivial technical complications, because one is no longer allowed to separate the small divisor problem plaguing the range equations from the implicit function problem represented by the bifurcation equations. The method we use is based on the analysis and resummation of the perturbation series through renormalisation group techniques [27,32,33,26]; for other renormalisation group approaches to small divisors problems in dynamical systems see for instance [12,39,43,38,40]. As in [19], the frequency vector of the perturbation will be assumed to satisfy the Bryuno condition; such a condition, originally introduced by Bryuno [14], has been studied recently in several small divisor problems arising in dynamical systems [32,33,34,40,43,49].…”

Reducibility of quasi-periodically forced circle flows

“…In this subsection, we construct a coordinate transformation U = U 1 such that U * f has no non-resonant modes. We construct this transformation using a homotopy method, which is different from the method used in [10,11,12,13,14,15]. Let ω ∈ R d , θ ∈ R, σ > 0, γ > 0, and y)) such that U 0 = I is the identity map, and…”

“…Renormalization transformations. Following [12,13,15], we express the Brjuno condition on ω (and, thus, on V) in terms of the summability of the series of numbers…”

“…Remark 2. The renormalization approach developed here is similar to, but technically more involved than, the renormalization approach to the construction of invariant tori for Hamiltonian and other vector fields [10,11,12,14,17,18], reducibility of skew-product flows on T d × SL(2, R) [15] and construction of lower-dimensional tori for Hamiltonian flows [13]. The small divisors encountered in these problems are produced by frequencies ν ∈ Z d that lie in the "resonant" regions, outside of certain "non-resonant" cones, surrounding some resonant planes, perpendicular to ω.…”

“…In the case of reducibility of skew-product flows on T d × SL(2, R), there is an additional resonant plane surrounded by frequencies corresponding to small divisors |ω •ν −2ρ|, where ±iρ are the eigenvalues of a matrix in the Lie algebra sl(2, R) [15]. In the case of lower-dimensional tori [13], there are finitely many resonant planes corresponding to small divisors |ω…”