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Renormalization of vector fields and Diophantine invariant tori

Published online by Cambridge University Press:  01 October 2008

HANS KOCH  and
SAŠA KOCIĆ
HANS KOCH
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, USA (email: koch@math.utexas.edu, kocic@math.utexas.edu)
SAŠA KOCIĆ
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, USA (email: koch@math.utexas.edu, kocic@math.utexas.edu)
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Abstract

We extend the renormalization group techniques that were developed originally for Hamiltonian flows to more general vector fields on 𝕋d×ℝℓ. Each Diophantine vector ω∈ℝd 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 ω1,ω2,…,ωd. Analogous manifolds for particular classes of vector fields (Hamiltonian, divergence-free, symmetric, reversible) are obtained simply by restricting 𝒲 to the corresponding subspace. We also discuss non-degeneracy conditions, and the resulting reduction in the number of parameters needed in parametrized families to guarantee the existence of invariant tori.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 28 , Issue 5 , October 2008 , pp. 1559 - 1585
Copyright
Copyright © 2008 Cambridge University Press

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