Renormalization of vector fields and Diophantine invariant tori

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0143385707000892 How to cite this article: HANS KOCH and SAŠA KOCIĆ (2008). Renormalization of vector elds and Diophantine invariant tori.Abstract. We extend the renormalization group techniques that were developed originally for Hamiltonian flows to more general vector fields on T d × R . Each Diophantine vector ω ∈ 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 ω 1 , ω 2 , . . . , ω d . 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.

Renormalization conjecture and rigidity theory for circle diffeomorphisms with breaks

“…The idea of renormalization originally came from statistical mechanics, where it provided an explanation for critical phenomena, by classifying systems into different universality classes, according to their scaling limits and corresponding critical exponents. In dynamics, apart from providing an explanation for the universality of infinitely renormalizable unimodal maps, renormalization methods have also led to advances in rigidity theory [4,5,[13][14][15][16][17][18][19]29], complex dynamics [27,30], KAM (Kolmogorov-Arnol'd-Moser) theory [20][21][22], break-up of invariant tori [1,25], and the reducibility of cocycles and skew-product flows [3,23].…”

A renormalization approach to lower-dimensional tori with Brjuno frequency vectors

“…In higher-dimensional problems, the question of the weakest possible condition is a fundamental open problem. Recently, we have developed a renormalization group approach to the construction of invariant tori with Brjuno frequency vectors for vector fields on T d × R m [16], using two general theorems, a normal form theorem and a stable manifold theorem, that we proved in [15]. The main objective of this paper is to extend the renormalization methods developed in [16] in order to construct analytic lower-dimensional tori of near-integrable Hamiltonian systems.…”

“…In [15], we constructed a renormalization scheme using a sequence of scaling matrices T ∈ SL(d, Z) generated by a multidimensional continued fraction algorithm introduced by Khanin, Lopes Dias and Marklof [11], and based on an algorithm by Lagarias [21]. The results of [11] provide good bounds in the case of Diophantine vectors ω, and allowed for the extension of the results of [17] for two degree of freedom Hamiltonians to higher dimensions [12].…”

“…As usual (see e.g. [15]), close to a non-degenerate Hamiltonian vector field, for which the q-component X q | p=0 is transversal to ω, one can further reduce the number of parameters needed to prove the existence of invariant tori by d − 1, by considering the translations in the p-variables.…”

Local conjugacy classes for analytic torus flows

“…However, in those works the renormalizations are based on the continued fraction expansion of irrational numbers. Here we are able to work on higher dimensions by making use of the multidimensional continued fractions algorithm introduced in [13] (see also [12,17,18]). This roughly corresponds to a flow on the homogeneous space SL(d, Z) \ SL(d, R) that, following Lagarias ideas [19], provides a strongly convergent continued fractions expansion for all vectors (see Section 2 below).…”