A note on the KAM theorem for partially integrable Hamiltonian systems

Action-angle variables and a KAM theorem for b-Poisson manifolds

In this article we prove an action-angle theorem for b-integrable systems on b-Poisson manifolds improving the action-angle theorem contained in [LMV11] for general Poisson manifolds in this setting. As an application, we prove a KAM-type theorem for b-Poisson manifolds.

奇异辛流形的作用-角坐标与KAM理论

Action-angle coordinates and KAM theory for singular symplectic manifolds

本专著研究$b^m$-Poisson流形上可积系统的分类与摄动问题，证明了其作用-角坐标定理与KAM定理，结合去奇异化技术推广得到折叠辛流形的KAM定理，还提出了带特异超曲面摄动的辛流形新KAM定理，为一般Poisson流形摄动理论研究奠定基础。

Reducing almost Lagrangian structures and almost CR geometries to partially integrable structures

This paper demostrates a method for analysing almost CR geometries $(H,J)$, by uniquley defining a partially integrable structure $(H,K)$ from the same data. Thus two almost CR geometries $(H,J)$ and $(H',J')$ are equivalent if and and only if they generate isomorphic induced partially integrable CR geometries $(H,K)$ and $(H',K')$, and if the set of CR morphisms between these spaces contains an element that maps $J$ to $J'$. Similar results hold for almost Lagrangian structures.

Closing lemma and KAM normal form

In this paper, we develop an approach to the problem of closing lemma based on KAM normal form. The new approach differs from existing $C^1$ perturbation approach and spectral approach, and can handle the high regularity, high dimensional cases and even Riemannian metric perturbations. Moreover, the proof is constructive and effective. We apply the method to the original nearly integrable setting of Poincaré and confirm several old and new conjectures with weak formulations. First, for Poincaré's original setting of nearly integrable systems, we prove that for typical perturbations, periodic o…

Nearly-integrable almost-symplectic Hamiltonian systems

Integrable Hamiltonian systems on almost-symplectic manifolds have recently drawn some attention. Under suitable properties, they have a structure analogous to those of standard symplectic-Hamiltonian completely integrable systems. Here we study small Hamiltonian perturbations of these systems. Preliminarily, we investigate some general properties of these systems. In particular, we show that if the perturbation is `strongly Hamiltonian' (namely, its Hamiltonian vector field is also a symmetry of the almost-Hamiltonian structure) then the system reduces, under an almost-symplectic version of s…

Uniqueness of Invariant Lagrangian Graphs in a Homology or a Cohomology Class

Given a smooth compact Riemannian manifold $M$ and a Hamiltonian $H$ on the cotangent space $T^*M$, strictly convex and superlinear in the momentum variables, we prove uniqueness of certain ergodic invariant Lagrangian graphs within a given homology or cohomology class. In particular, in the context of quasi-integrable Hamiltonian systems, our result implies global uniqueness of Lagrangian KAM tori with rotation vector $ρ$. This result extends generically to the $C^0$-closure of KAM tori.