06-269 Hans Koch and Sasa Kocic

A renormalization group approach to quasiperiodic motion with Brjuno frequencies (78K, plain TeX) Sep 27, 06

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Abstract. We introduce a renormalization group scheme that applies to vector fields on $torus^d\times\real^m$ with frequency vectors that satisfy a Brjuno condition. Earlier approaches were restricted to Diophantine frequencies, due to a limited control of multidimensional continued fractions. We get around this restriction by avoiding the use of a continued fractions expansion. Our results concerning invariant tori generalize those of reference [17] from Diophantine to Brjuno type frequency vectors. In particular, each Brjuno 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 $\omega_1,\omega_2,\ldots,\omega_d$.

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