Periodic orbits of Hamiltonian systems and symplectic topology of coisotropic submanifolds

本提案侧重于与国际和平倡议以前的工作密切相关的两个项目。第一组问题涉及到所谓的Conley猜想和周期轨道的几乎存在定理。Conley猜想的一般形式证明了辛非球闭流形的哈密顿微分同态存在无穷多个周期点。这一猜想最近被Hingston和PI.证实。然而，这个问题的许多方面还需要进一步研究。例如，即使流形不是非球面的，猜想也会成立，但哈密顿微分同态有足够多的不动点。这项拟议的研究解决了康利猜想的这一点和其他一些方面。几乎存在定理保证了对于一大类辛流形，在适当的自治哈密顿量的几乎所有水平上都存在周期轨道。这一定理在概念和技术层面上都与康利猜想和温斯坦猜想密切相关。提案中描述的一个项目旨在通过证明没有周期轨道的能量值集合在任何地方都不稠密来补充几乎存在定理。第二部分讨论了余迷向子流形的辛拓扑性质。这些性质推广了拉格朗日交性质和Maslov类刚性，在动力学中有重要的应用。此外，出现了一幅普遍的图景，使得人们能够将不存在精确的拉格朗日嵌入和接触型超曲面上存在闭特征等事实视为一种现象的特例。最近由PI启动并在提案中概述的计划的主要目标是进一步分析和扩展这幅图。哈密顿动力系统描述了许多类可以忽略耗散力的物理过程。例如，天体力学中的行星运动和一些电磁或磁动力学过程可以而且通常被视为哈密顿动力系统。周期轨道(即循环运动)的研究是现代哈密顿动力系统理论和辛几何的核心问题之一。周期轨道是无处不在的：绝大多数哈密顿系统都有周期轨道，对于一大类系统来说，不同的周期轨道的数量是无限的。对这一现象的分析是以国际和平研究所最近的工作为基础的，是拟议研究的主要目标之一。所讨论的这类动力系统包括描述电荷在磁场中运动的那些系统，所提出的研究在物理和力学的数学方面具有潜在的应用。

The present proposal focuses on two projects closely related to the PI's previous work. The first group of problems addressed concerns the so-called Conley conjecture and the almost existence theorem for periodic orbits. The general form of the Conley conjecture asserts the existence of infinitely many periodic points of a Hamiltonian diffeomorphism of a symplectically aspherical, closed manifold. This conjecture has recently been established by Hingston and the PI.However, many aspects of the problem require further investigation. For instance, one can expect the conjecture to hold even when the manifold is not aspherical, but the Hamiltonian diffeomorphism has sufficiently many fixed points. The proposed research addresses this and some other aspects of the Conley conjecture. The almost existence theorem guarantees the existence of periodic orbits on almost all levels of a proper, autonomous Hamiltonian for a broad class of symplectic manifolds. This theorem is closely related, on both conceptual and technical levels, to the Conley conjecture and the Weinstein conjecture. A project described in the proposal aims at complementing the almost existence theorem by showing that the set of energy values without periodic orbits is nowhere dense. The second part of the proposal focuses on symplectic topological properties of coisotropic submanifolds. These properties generalize the Lagrangian intersection property and the Maslov class rigidity and have important application in dynamics. Moreover, a general picture is emerging, enabling one to treat such facts as non-existence of exact Lagrangian embeddings and the existence of closed characteristics on a contact type hypersurface as particular cases of one phenomenon. The main goal of the program started recently by the PI and outlined in the proposal is to further analyze and extend this picture.Hamiltonian dynamical systems describe many classes of physical processes in which dissipative forces can be neglected. For example, planetary motion in celestial mechanics and some electro- or magneto-dynamical processes can be, and usually are, treated as Hamiltonian dynamical systems. One of the classical subjects lying at the very core of modern theory of Hamiltonian dynamical systems and symplectic geometry is the study of periodic orbits (i.e., cyclic motions). Periodic orbits are ubiquitous: a vast majority of Hamiltonian systems have periodic orbits and the number of distinct periodic orbits is infinite for a broad class of systems. The analysis of this phenomenon, building on the PI's recent work, is among the main objectives of the proposed research. The class of dynamical systems in question includes those describing the motion of a charge in a magnetic field and the proposed research has potential applications to physics and mathematical aspects of mechanics.