A renormalization group approach to quasiperiodic motion with Brjuno frequencies

作者：Koch Hans^*; Kocic Sasa

来源：Ergodic Theory and Dynamical Systems, 2010, 30(04): 1131-1146.

DOI：10.1017/S014338570900042X

We introduce a renormalization group scheme that applies to vector fields on T(d) x R(m) 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. Kocic, 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 omega is an element of R(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) , ... , omega(d).

• 出版日期2010-8