On the topology of nearly-integrable Hamiltonians at simple resonances

Abstract

We show that, in general, averaging at simple resonances a real-analytic, nearly-integrable Hamiltonian, one obtains a one-dimensional system with a cosine-like potential; ‘in general’ means for a generic class of holomorphic perturbations and apart from a finite number of simple resonances with small Fourier modes; ‘cosine-like’ means that the potential depends only on the resonant angle, with respect to which it is a Morse function with one maximum and one minimum. Furthermore, the (full) transformed Hamiltonian is the sum of an effective one-dimensional Hamiltonian (which is, in turn, the sum of the unperturbed Hamiltonian plus the cosine-like potential) and a perturbation, which is uniformly exponentially small. As a corollary, under the above hypotheses, if the unperturbed Hamiltonian is also strictly convex, the effective Hamiltonian at any simple resonance (apart a finite number of low-mode resonances) has the phase portrait of a pendulum. The results presented in this paper are an essential step in the proof (in the ‘mechanical’ case) of a conjecture by Arnold–Kozlov–Neishdadt [Arnold V I, Kozlov V V and Neishtadt A I 2006 Mathematical aspects of classical and celestial mechanics Encyclopaedia of Mathematical Sciences 3rd edn vol 3 (Berlin: Springer), remark 6.8, p 285], claiming that the measure of the ‘non-torus set’ in general nearly-integrable Hamiltonian systems has the same size of the perturbation; compare [Biasco L and Chierchia L 2015 On the measure of Lagrangian invariant tori in nearly-integrable mechanical systems Rendiconti Lincei. Mat. Appl. 26 1–10 and Biasco L and Chierchia L KAM Theory for Secondary Tori (arXiv:1702.06480v1 [math.DS])].