The KAM theorem on the modulus of continuity about parameters

In this paper, we study the Hamiltonian systems H(y, x, xi, epsilon) = <omega(xi),y >+epsilon P(y, x, xi, epsilon), where omega and P are continuous about xi. We prove that persistent invariant tori possess the same frequency as the unperturbed tori, under a certain transversality condition and a weak convexity condition for the frequency mapping omega. As a direct application, we prove a Kolmogorov-Arnold-Moser (KAM) theorem when the perturbation P holds arbitrary Holder continuity with respect to the 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 the parameter beyond Holder's type.