Periodic orbits of Hamiltonian systems and symplectic topology of coisotropic submanifolds

[摘要]获奖：dms -1007149首席研究员：Viktor ginzburg本提案主要关注与PI先前工作密切相关的几个项目。第一组问题涉及到康利猜想的推广。这个猜想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。Conley猜想由Hingston（对于环面）、PI建立，并最终推广到所有零陈氏类的辛流形。然而，这个问题的许多方面还需要进一步调查。本文给出了一种证明Conley猜想的方法，该方法适用于具有极大极小Chern类的流形，以及具有“太多”不动点的某些类型的辛同态和哈密顿微分同态。对哈密顿系统周期轨道的概存在定理进行了改进，并对与康利猜想密切相关的扭曲测地线流的周期轨道进行了研究。第二部分着重讨论了各向同性子流形的辛拓扑性质。这些性质将拉格朗日交点性质以及Liouville和Maslov类刚性推广到某一类共同性子流形，即所谓的稳定子流形。此外，一个一般的图景已经出现，使人们能够将诸如不存在精确拉格朗日嵌入和在接触型超表面上存在闭合特征等事实视为一种现象的特殊情况。由PI启动并在提案中继续的计划的主要目标是进一步分析这张图，并将共同性刚性结果扩展到更广泛的子流形类别。哈密顿动力系统描述了许多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动力学或磁动力学过程可以而且通常被视为哈密顿动力学系统。哈密顿动力系统和辛几何的现代理论的核心之一是周期轨道（即循环运动）的研究。周期轨道是无处不在的：绝大多数的哈密顿系统都有周期轨道，而且对于大多数系统来说，不同周期轨道的数量是无限的。在PI最近工作的基础上对这一现象进行分析，是拟议研究的主要目标之一。例如，PI提出要证明某种类型的哈密顿系统有无限多个周期轨道。所讨论的动力系统类别包括那些描述磁场中电荷运动的系统，所提出的研究在力学的物理和数学方面具有潜在的应用。

AbstractAward: DMS-1007149Principal Investigator: Viktor GinzburgThe present proposal focuses on several projects closely related to the PI's previous work. The first group of problems addressed concerns generalizations of the Conley conjecture. Thisconjecture asserts the existence of infinitely many periodic points of a Hamiltonian diffeomorphism of a symplectically aspherical, closed manifold. The Conley conjecture has been established by Hingston (for tori), the PI and eventually generalized to all symplectic manifolds with zero Chern class. However, many aspects of the problem require further investigation. The PI outlines an approach to the proof of the Conley conjecture for manifolds with large minimal Chern class and to some classes of symplectomorphisms and Hamiltonian diffeomorphisms with "too many" fixed points. A refinement of the almost existence theorem for periodic orbits of Hamiltonian systems and an investigation of periodic orbits of twisted geodesic flows, closely related to the Conley conjecture, are also considered in the proposal. The second part of the proposal focuses on symplectic topological properties of coisotropic submanifolds. These properties generalize the Lagrangian intersection property and the Liouville and Maslov class rigidity to a certain class of coisotropic submanifolds, the so-called stable submanifolds. Moreover, a general picture has emerged, 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 by the PI and continued in the proposal is to further analyze this picture and to extend coisotropic rigidity results to a broader class of submanifolds.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. For instance, the PI proposes to show that Hamiltonian systems of a certain type have infinitely many periodic orbits. 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.