06-220 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, 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.

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