KAM theorem on modulus of continuity about parameter

TL;DR

This paper extends the KAM theorem to Hamiltonian systems with parameters that are only continuous, not necessarily H"older continuous, demonstrating persistence of invariant tori under weak regularity conditions.

Contribution

It introduces a novel approach to KAM theory for systems with parameters of mere continuity, beyond traditional H"older assumptions.

Findings

Proves persistence of invariant tori with the same frequency under weak conditions.

Establishes a KAM theorem for perturbations with arbitrary H"older continuity.

Extends results to infinite-dimensional Hamiltonian systems.

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.