# Towards continuity: Universal frequency-preserving KAM persistence and remaining regularity

## TL;DR

This paper establishes a universal frequency-preserving KAM theorem for finitely differentiable Hamiltonian systems, demonstrating persistence and regularity of invariant tori under critical smoothness conditions, with explicit non-Hölder regularity analysis.

## Contribution

It introduces a novel KAM theorem with sharp differentiability conditions that preserves universal frequencies and handles critical smoothness, expanding the scope of KAM theory.

## Key findings

- Proves persistence of invariant tori with universal Diophantine frequencies.
- Demonstrates non-Hölder regularity of invariant tori and conjugation.
- Provides new systems where previous KAM methods fail but this approach succeeds.

## Abstract

Beyond H\"{o}lder'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-H\"older 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.