
University of Helsinki, Helsinki 2006 Renormalization methods in KAM theoryEmiliano De SimoneDoctoral dissertation, May 2006. It is well known that an integrable (in the sense of ArnoldJost) Hamiltonian system gives rise to quasiperiodic motion with trajectories running on invariant tori. These tori foliate the whole phase space. If we perturb an integrable system, the KolmogorowArnoldMoser (KAM) theorem states that, provided some nondegeneracy condition and that the perturbation is sufficiently small, most of the invariant tori carrying quasiperiodic motion persist, getting only slightly deformed. The measure of the persisting invariant tori is large together with the inverse of the size of the perturbation. In the first part of the thesis we shall use a Renormalization Group (RG) scheme in order to prove the classical KAM result in the case of a non analytic perturbation (the latter will only be assumed to have continuous derivatives up to a sufficiently large order). We shall proceed by solving a sequence of problems in which theperturbations are analytic approximations of the original one. We will finally show that the approximate solutions will converge to a differentiable solution of our original problem. In the second part we will use an RG scheme using continuous scales, so that instead of solving an iterative equation as in the classical RG KAM, we will end up solving a partial differential equation. This will allow us to reduce the complications of treating a sequence of iterative equations to the use of the Banach fixed point theorem in a suitable Banach space. The title page of the publication
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