Kolmogorov's Theorem for Degenerate Hamiltonian Systems with Continuous Parameters

Abstract

In this paper, we study Kolmogorov type theorems for small perturbations of certain degenerate Hamiltonian systems, indexed by a parameter ξ, with H(y, x, ξ) = ω(ξ), y + εP(y, x, ξ, ε), where ε > 0. We assume the frequency map ω is continuous about ξ, and the perturbation function P(y, x, •, ε) is Hölder continuous about ξ. We prove that persistent invariant tori possess the same frequency as the unperturbed tori, under certain topological degree condition and weak convexity condition for the frequency mapping. To our knowledge, this is the first results about KAM theorem under similar settings, with only Hölder continuous dependence of P on the parameter.

An infinite dimensional KAM theorem with normal degeneracy

“…See (A0) in section 1.3 for details. It should be pointed out that the topological degree can also be used to study frequency-preserving in the KAM theory, see [5,15,19].…”

Towards continuity: Universal frequency-preserving KAM persistence and remaining regularity

“…Considering Diophantine nonresonance, the invariant tori are shown in classical KAM theorems to be preserved under analytic settings. It should be pointed out that, if a constant Diophantine frequency is prescribed in advance, one could obtain KAM persistence with frequency being unchanged by proposing certain nondegeneracy or transversality conditions, which preserves more dynamics from the perturbed one, see Salamon [35], Du et al [11] and Tong et al [38] for instance, respectively, and this analytic torus is interestingly shown to be never isolated by Eliasson et al in [13]. Therefore, beyond analyticity, it is still interesting to touch the minimum initial regularity for Hamiltonian to which KAM could applied.…”