Kolmogorov's Theorem for Degenerate Hamiltonian Systems with Continuous Parameters

Abstract

In this paper, we investigate Kolmogorov type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter $\xi$ as \( H(y,x,ξ) = \langleω(ξ),y\rangle + \varepsilon P(y,x,ξ,\varepsilon) \) where $\epsilon >0$. We assume that the frequency map, $\omega$, is continuous with respect to $\xi$. Additionally, the perturbation function, $P(y,x,\cdot ,\epsilon )$, maintains Hölder continuity about $\xi$. We prove that persistent invariant tori retain the same frequency as those of the unperturbed tori, under certain topological degree conditions and a weak convexity condition for the frequency mapping. Notably, this paper presents, to our understanding, pioneering results on the KAM theorem under such conditions-with only assumption of continuous dependence of frequency mapping $\omega$ on the parameter.

31 pages

Cited by in corpus (1)

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