Periodic orbits of Hamiltonian systems and symplectic topology of coisotropic submanifolds

本提案重点关注与 PI 之前工作密切相关的两个项目。 第一组解决的问题涉及所谓的康利猜想和周期轨道的几乎存在定理。康利猜想的一般形式断言辛非球面闭流形的哈密顿微分同胚存在无限多个周期点。这个猜想最近被 Hingston 和 PI 证实了。但是，问题的许多方面还需要进一步调查。例如，即使流形不是非球面，但哈密顿微分同胚有足够多的不动点，我们也可以预期猜想成立。拟议的研究解决了康利猜想的这个问题和其他一些方面。几乎存在定理保证了一大类辛流形的适当的、自主的哈密顿量的几乎所有级别上都存在周期轨道。 该定理在概念和技术层面上都与康利猜想和韦恩斯坦猜想密切相关。提案中描述的一个项目旨在通过证明没有周期性轨道的能量值集在任何地方都不稠密来补充几乎存在定理。该提案的第二部分重点关注各向同性子流形的辛拓扑性质。这些性质概括了拉格朗日交集性质和马斯洛夫级刚度，在动力学中具有重要应用。 此外，一幅总体图景正在出现，使人们能够将诸如不存在精确拉格朗日嵌入以及接触型超曲面上存在闭合特征等事实视为一种现象的特殊情况。 PI 最近启动并在提案中概述的该计划的主要目标是进一步分析和扩展这一图景。哈密尔顿动力系统描述了许多类物理过程，其中耗散力可以忽略不计。例如，天体力学中的行星运动和一些电动力或磁动力过程可以并且通常被视为哈密顿动力系统。哈密​​顿动力系统和辛几何现代理论的核心经典学科之一是对周期轨道（即循环运动）的研究。周期轨道无处不在：绝大多数哈密顿系统都具有周期轨道，并且对于一大类系统来说，不同周期轨道的数量是无限的。基于 PI 最近的工作，对这一现象的分析是拟议研究的主要目标之一。所讨论的动力系统类别包括描述磁场中电荷运动的动力系统，所提出的研究在力学的物理和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.