Universal frequency-preserving KAM persistence via modulus of continuity

TL;DR

This paper extends KAM theory to finite smoothness contexts with modulus of continuity, proving persistence of invariant tori with sharp differentiability conditions and preserving universal Diophantine frequencies.

Contribution

It generalizes classical KAM results by incorporating modulus of continuity and extends Jackson approximation to this setting, establishing a new regularity theorem.

Findings

Persistence of KAM tori with universal Diophantine frequencies.

Extension of Jackson approximation theorem to modulus of continuity.

Establishment of a KAM theorem with sharp differentiability hypotheses.

Abstract

In this paper, we study the persistence and remaining regularity of KAM invariant torus under sufficiently small perturbations of a Hamiltonian function together with its derivatives, in sense of finite smoothness with modulus of continuity, as a generalization of classical H\"{o}lder continuous circumstances. To achieve this goal, we extend the Jackson approximation theorem to the case of modulus of continuity, and establish a corresponding regularity theorem adapting to the new iterative scheme. Via these tools, we establish a KAM theorem with sharp differentiability hypotheses, which asserts that the persistent torus keeps prescribed universal Diophantine frequency unchanged and reaches the regularity for persistent KAM torus beyond H\"older's type.