Hamilton-Arnold Chord And Periodic Orbits

On the strict Arnold chord property and coisotropic submanifolds of complex projective space

Let $α$ be a contact form on a manifold $M$, and $L\subseteq M$ a closed Legendrian submanifold. I prove that $L$ intersects some characteristic for $α$ at least twice if all characteristics are closed and of the same period, and $α$ embeds nicely into the product of $\mathbb{R}^{2n}$ and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most $π/2$. This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.

接触瞬子与Weinstein猜想及Arnold弦猜想的证明

Contact instantons and Proofs of Weinstein's conjecture and Arnold's chord conjecture

本文基于接触Hamilton动力学与接触瞬子模空间理论，通过利用Legendrian接触瞬子上同调的基本类存在性、评估横截性以及接触同胚的Legendrian化几何构造，完整证明了Weinstein猜想和Arnold弦猜想。

Holomorphic Disks and the Chord Conjecture

We prove (a weak version of) Arnold's Chord Conjecture using Gromov's ``classical'' idea in to produce holomorphic disks with boundary on a Lagrangian submanifold.

关于$\boldsymbol{\tau^{2m}\times\boldsymbol{\tau}\boldsymbol{P}^n}$上的退化Arnold猜想

On the degenerate Arnold conjecture on $\mathbb T^{2m}\times \mathbb C\mathbb P^n$

针对闭连通辛流形哈密顿微分同胚可缩不动点数量的Arnold猜想，本文基于作者与Izydorek对CPⁿ的研究方法，证明当m,n≥1时，T²ᵐ×CPⁿ上在CPⁿ方向C⁰接近恒等的哈密顿微分同胚至少有2m+n+1个可缩不动点。

Periodic orbits in the restricted three-body problem and Arnold's $J^+$-invariant

We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold's $J^+$-invariant, we introduce invariants of periodic orbits in planar Stark-Zeeman systems and study their behaviour.

退化哈密顿量的周期轨道估计

Estimate of Periodic Orbits of Degenerate Hamiltonians

本文综述退化Arnold猜想的研究进展，提出结合Floer上同调上的量子同调作用与通用能量估计的新证明思路，在Floer理论框架下证明某些特殊情形下的退化Arnold猜想。

An iterated graph construction and periodic orbits of Hamiltonian delay equations

According to the Arnold conjectures and Floer's proofs, there are non-trivial lower bounds for the number of periodic solutions of Hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres. We use an iterated graph construction and Lagrangian Floer homology to show that these lower bounds also hold for certain Hamiltonian delay equations.

The $E$-cohomological Conley Index, Cup-Lengths and the Arnold Conjecture on $T^{2n}$

We give a new proof of the strong Arnold conjecture for $1$-periodic solutions of Hamiltonian systems on tori, that was first shown by C. Conley and E. Zehnder in 1983. Our proof uses other methods and is shorter than the previous one. We first show that the $E$-cohomological Conley index, that was introduced by the first author recently, has a natural module structure. This yields a new cup-length and a lower bound for the number of critical points of functionals. Then an existence result for the $E$-cohomological Conley index, which applies to the setting of the Arnold conjecture, paves the …

奇异辛流形的 Arnold 猜想

The Arnold conjecture for singular symplectic manifolds

本文通过把奇异辛结构关联到光滑辛形式，证明多类 $b^m$-辛流形的 Arnold 猜想，并给出依赖拓扑的周期 Hamilton 轨道数下界。

Existence of noncontractible periodic orbits of Hamiltonian system separating two Lagrangian tori on $T^*\T^n$ with application to non convex Hamiltonian systems

In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on $T^*\T^n$ separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in \cite{P} in a sharp way. As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on $T^*\T^2$. This solves a problem of Arnold in \cite{A}.

Arnold-Givental猜想的一个证明

A proof of the Arnold-Givental conjecture

该研究完整证明了Arnold-Givental猜想，结合积分Floer理论、哈密顿Floer上同调约化及定制的$\mathbb Z/2$-等变Floer理论局部化新思路，验证了闭辛流形中横截相交的不动点集计数下界不等式。

Floer homology for negative line bundles and Reeb chords in pre-quantization spaces

In this article we prove existence of Reeb orbits for Bohr-Sommerfeld Legendrians in certain pre-quantization spaces. We give a quantitative estimate from below. These estimates are obtained by studying Floer homology for fibre-wise quadratic Hamiltonian functions on negative line bundles.

利用微局部层理论估计Reeb弦

Estimating Reeb chords using microlocal sheaf theory

针对1-射流丛中闭Legendrian子流形的Reeb弦估计问题，本文结合微局部层理论构建对偶正合三角并利用持续结构，给出了Reeb弦数量及与哈密顿推挤间Reeb弦数量的Betti数下界。

The Calabi invariant and the least number of periodic solutions of locally Hamiltonian equations

In this paper we prove a lower bound for the least number of one-periodic solutions of nondegenerate locally Hamiltonian equations on compact symplectic manifolds in terms of the Betti numbers of the Novikov homology associated to the Calabi invariant of the locally Hamiltonian equations. Our result improves lower bounds obtained by Lê-Ono and Ono for the least number of nondegenerate locally Hamiltonian symplectic fixed points. Our result also generalizes the homological Arnold conjecture proved by Fukaya-Ono and Liu-Tian.

Vortex sheets and diffeomorphism groupoids

In 1966 V.Arnold suggested a group-theoretic approach to ideal hydrodynamics in which the motion of an inviscid incompressible fluid is described as the geodesic flow of the right-invariant $L^2$-metric on the group of volume-preserving diffeomorphisms of the flow domain. Here we propose geodesic, group-theoretic, and Hamiltonian frameworks to include fluid flows with vortex sheets. It turns out that the corresponding dynamics is related to a certain groupoid of pairs of volume-preserving diffeomorphisms with common interface. We also develop a general framework for Euler-Arnold equations for …