Moser's Theorem with Frequency-preserving

TL;DR

This paper proves a new version of Moser's theorem ensuring the persistence of invariant tori with preserved frequencies under small perturbations, even with minimal regularity assumptions on the mappings.

Contribution

It introduces the first approach to Moser's theorem with frequency-preserving, accommodating continuous frequency mappings and perturbations beyond Lipschitz or H"older regularity.

Findings

Invariant tori persist under small perturbations with frequency-preserving.

Persistence holds even with continuous, not necessarily Lipschitz or H"older, mappings.

The results apply without dimension restrictions on parameters.

Abstract

This paper mainly concerns the KAM persistence of the mapping $\mathscr{F}:\mathbb{T}^{n}\times E\rightarrow \mathbb{T}^{n}\times \mathbb{R}^{n}$ with intersection property, where $E\subset \mathbb{R}^{n}$ is a connected closed bounded domain with interior points. By assuming that the frequency mapping satisfies certain topological degree condition and weak convexity condition, we prove some Moser type results about the invariant torus of mapping $\mathscr{F}$ with frequency-preserving under small perturbations. To our knowledge, this is the first approach to Moser's theorem with frequency-preserving. Moreover, given perturbed mappings over $ \mathbb{T}^n $, it is shown that such persistence still holds when the frequency mapping and perturbations are only continuous about parameter beyond Lipschitz or even H\"older type. We also touch the parameter without dimension limitation problem under such settings.