 06220 Hans Koch and Sasa Kocic
 Renormalization of Vector Fields
and Diophantine Invariant Tori
(116K, plain TeX)
Aug 11, 06

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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, divergencefree,
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.
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