Kolmogorov's Theorem for Degenerate Hamiltonian Systems with Continuous Parameters

In this paper, we investigate Kolmogorov type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter $\xi$ as $H(y,x,\xi) = \langle\omega(\xi),y\rangle + \varepsilon P(y,x,\xi,\varepsilon)$ where $\varepsilon>0$. We assume that the frequency map, $\omega$, is continuous with respect to $\xi$. Additionally, the perturbation function, $P(y,x,\cdot, \varepsilon)$, maintains H\"{o}lder 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.