PFH谱不变量与$C^\infty$闭合引理

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…

辛动力学中的强闭合引理

Strong closing lemmas in Hamiltonian dynamics

本文探讨了利用谱不变量证明的辛动力学中强闭合引理，涵盖低维和高维情况，以及与极小超曲面的类比结果。

Periodic Floer homology and the smooth closing lemma for area-preserving surface diffeomorphisms

We prove a very general Weyl-type law for Periodic Floer Homology, estimating the action of twisted Periodic Floer Homology classes over essentially any coefficient ring in terms of the grading and the degree, and recovering the Calabi invariant of Hamiltonians in the limit. We also prove a strong non-vanishing result, showing that under a monotonicity assumption which holds for a dense set of maps, the Periodic Floer Homology has infinite rank. An application of these results yields that a $C^{\infty}$-generic area-preserving diffeomorphism of a closed surface has a dense set of periodic poin…

穿孔曲面上保面积映射周期轨道的通有稠密性

Generic density of periodic orbits of area-preserving maps on punctured surfaces

本文在有限穿孔闭曲面上证明保面积微分同胚的C∞闭合引理，并由此得到通有周期点稠密性，进而在穿孔2-球面上建立周期轨道均匀分布的定量结果。

环面Floer谱不变量、对称积链接与韦尔定律

Orbifold Floer spectral invariants, symmetric product links and Weyl laws

本文提出利用对称积 orbifold 的谱不变量证明辛流形 Hamilton 算子的光滑闭合引理，并通过 2 球面的示例展示方法，同时指出高维中构造拉格朗日链接仍是难题。

Dense existence of periodic Reeb orbits and ECH spectral invariants

In this paper, we prove (1): for any closed contact three-manifold with a $C^\infty$-generic contact form, the union of periodic Reeb orbits is dense, (2): for any closed surface with a $C^\infty$-generic Riemannian metric, the union of closed geodesics is dense. The key observation is $C^\infty$-closing lemma for 3D Reeb flows, which follows from the fact that the embedded contact homology (ECH) spectral invariants recover the volume.

KAM 分裂与稳定超曲面的等分布周期轨道

KAM splittings and equidistributed periodic orbits for stable hypersurfaces

本文证明辛4流形上满足同调条件的稳定超曲面可被闭特征等分布的超曲面C∞逼近，结合KAM理论与定量闭合引理，并推广到三维流形上的体积保持向量场。