KAM theorem on modulus of continuity about parameter

Abstract

In this paper, we study the Hamiltonian systems $ H\left( {y,x,\xi ,\varepsilon } \right) = \left\langle {\omega \left( \xi \right),y} \right\rangle + \varepsilon P\left( {y,x,\xi ,\varepsilon } \right) $, where $ \omega $ and $ P $ are continuous about $ \xi $. We prove that persistent invariant tori possess the same frequency as the unperturbed tori, under certain transversality condition and weak convexity condition for the frequency mapping $ \omega $. As a direct application, we prove a KAM theorem when the perturbation $P$ holds arbitrary H\"{o}lder continuity with respect to parameter $ \xi $. The infinite dimensional case is also considered. To our knowledge, this is the first approach to the systems with the only continuity in parameter beyond H\"older's type.