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A renormalization group approach to quasiperiodic motion with Brjuno frequencies

Published online by Cambridge University Press:  17 July 2009

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 introduce a renormalization group scheme that applies to vector fields on 𝕋d×ℝm with frequency vectors that satisfy a Brjuno condition. Earlier approaches were restricted to Diophantine frequencies, owing 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 [H. Koch and S. Kocić, Renormalization of vector fields and Diophantine invariant tori. Ergod. Th. & Dynam. Sys.28 (2008), 1559–1585] from Diophantine- to Brjuno-type frequency vectors. In particular, each Brjuno 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 4 , August 2010 , pp. 1131 - 1146
Copyright
Copyright © Cambridge University Press 2009

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