Towards continuity: Universal frequency-preserving KAM persistence and remaining regularity
Zhicheng Tong, Yong Li
TL;DR
The paper advances KAM theory for finitely differentiable Hamiltonians by introducing a modulus-of-continuity framework that enables universal frequency-preserving persistence beyond Hölder regularity. It develops a Jackson-type approximation suitable for general moduli and an abstract regularity-iteration to track non-Hölder smoothness of the invariant torus and the conjugation, under a generalized Dini-type condition. The main contribution is a frequency-preserving KAM theorem with sharp differentiability hypotheses, plus explicit remaining regularity results under Hölder, Log Hölder, and GLH-type continuities, including two explicit Hamiltonians that fall outside classical Hölder theory. This approach broadens KAM applicability to continuous-type regularity, clarifies the relation between Diophantine nonresonance and residual smoothness, and yields concrete constructions where prior theorems could not apply.
Abstract
Beyond Hölder's type, this paper mainly concerns the persistence and remaining regularity of an individual frequency-preserving KAM torus in a finitely differentiable Hamiltonian system, even allows the non-integrable part being critical finitely smooth. To achieve this goal, besides investigating the Jackson approximation theorem towards only modulus of continuity, we demonstrate an abstract regularity theorem adapting to the new iterative scheme. Via these tools, we obtain a KAM theorem with sharp differentiability hypotheses, asserting that the persistent torus keeps prescribed universal Diophantine frequency unchanged. Further, the non-Hölder regularity for invariant KAM torus as well as the conjugation is explicitly shown by introducing asymptotic analysis. To our knowledge, this is the first approach to KAM on these aspects in a continuous sense, and we also provide two systems, which cannot be studied by previous KAM but by ours.