Hans Koch and Sasa Kocic
Renormalization of Vector Fields
and Diophantine Invariant Tori
(116K, plain TeX)
ABSTRACT. We extend the renormalization group techniques that were
developed originally for Hamiltonian flows to more general vector fields
on $\torus^d\times\real^\ell$. Each Diophantine vector $\omega\in\real^d$
determines an analytic manifold $W$ of infinitely renormalizable vector fields,
and each vector field on $W$ is shown to have an elliptic invariant $d$-torus
with frequencies $\omega_1,\omega_2,\ldots,\omega_d$. Analogous manifolds
for particular classes of vector fields (Hamiltonian, divergence-free,
symmetric, reversible) are obtained simply by restricting $\WW$ to the
corresponding subspace. We also discuss nondegeneracy conditions, and the
resulting reduction in the number of parameters needed in parametrized
families to guarantee the existence of invariant tori.