Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori

Abstract

We construct a rigorous renormalization scheme for two degree-of-freedom analytic Hamiltonians, associated with Diophantine frequency vectors. We prove the existence of an attracting, integrable limit set of the renormalization. As an application of this renormalization scheme, we give a proof of a KAM theorem. We construct analytic invariant tori with Diophantine frequency vectors for near-integrable Hamiltonians in the domain of attraction of this limit set.

Renormalization of vector fields

“…A construction along these lines, of analytic invariant tori with self-similar frequency vectors, can be found in [74]. In the case d = 2, the procedure was generalized by Kocić [81] to frequency vectors ω that satisfy a Diophantine condition (with restrictions on the Diophantine exponent), using a sequence of RG transformation R n associated with the continued fraction expansion of the rotation number ω 1 /ω 2 . Recent work by Marklov, Lopes Dias, and Khanin [70] extends this construction to arbitrary Diophantine vectors ω in dimensions d ≥ 2, after introducing an appropriate multidimensional continued fraction expansion.…”

“…This requires at least r ≥ 1. Some recent work [77,81,78] follows a different approach, based entirely on semiconjugacies, where it is natural to use equation (5.13) directly. So there is no need to work with differentiable functions.…”

“…This argument uses specific properties of the fixed point H * and of its scaling map Λ H * . The other steps in the construction, which we discuss first, can also be adapted to the study of near-integrable Hamiltonians [81]. Notice that the RG transformation used near H * is also of the form (5.7), but with…”

“…We will describe some of the ideas below (with slight modifications, to synchronize the definitions with those used in previous sections). Presumably, the method can be generalized to Diophantine frequencies, using sequences of RG transformations as in [95,81].…”

“…[10]. A renormalization group analysis of the type described above was carried out in [81]. It is shown that if ω 0 is Diophantine, with β < √ 2 − 1, then H n − K n → 0 in a space of analytic Hamiltonians, provided that H 0 is sufficiently close to K 0 .…”