KAM theorem on modulus of continuity about parameter

In this paper, we study the Hamiltonian systems $H\left(y,x,\xi ,\epsilon \right)=⟨\omega \left(\xi \right),y⟩+\epsilon P\left(y,x,\xi ,\epsilon \right)$, 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 Hölder 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 Hölder’s type.