Periodic orbits of Hamiltonian systems

该提案主要关注哈密顿系统周期轨道的存在性问题以及与PI先前工作密切相关的其他几个问题。在建议中考虑的第一组问题涉及康利猜想的推广。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）和PI建立，并最终推广到所有具有零陈氏类的闭辛流形和负单调辛流形。PI对这些结果提出了各种进一步的推广，并提出了一个程序来研究已知猜想不成立的情况。特别地，PI将研究具有双曲不动点的哈密顿微分同态的无穷多个周期轨道的存在性，这是Franks？大最小陈氏数流形的Conley猜想。该提案的第二部分涉及类似的关于里布流周期性轨道的问题。最后，本文最后一部分的主题是研究具有有限多周期轨道的哈密顿系统及其作用和指标上的某些同调共振关系。这些项目的最终目标是更好地理解哈密顿系统周期轨道丰富的潜在原因。PI使用的方法本质上是花理论。一个重要的新成分来自PI？s最近与Gurel共同研究了具有双曲不动点的哈密顿微分同态的周期轨道。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。对这种现象的分析是本研究的主要目标之一。提出的研究的一个新成分是哈密顿系统的局部和全局特征之间的相互作用的新形式。例如，PI提出要证明具有某种局部行为的哈密顿系统必须有无限多个周期轨道。提出的研究在力学的物理和数学方面有潜在的应用。

The proposal focuses on the existence problem for periodic orbits of Hamiltonian systems and several other questions closely related to the PI's previous work. The first group of problems considered in the proposal concerns generalizations of the Conley conjecture. This conjecture 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) and the PI, and eventually generalized to all closed symplectic manifolds with zero Chern class and to negative monotone symplectic manifolds. The PI proposes a variety of further generalizations of these results as well as a program to investigate the cases where the conjecture is known not to hold. In particular, the PI will study the existence of infinitely many periodic orbits for Hamiltonian diffeomorphisms with a hyperbolic fixed point, higher dimensional generalizations of Franks? theorem, and the Conley conjecture for manifolds with large minimal Chern number. The second part of the proposal concerns a similar circle of questions for periodic orbits of Reeb flows. Finally, the main subject of the last part of the proposal is a study of Hamiltonian systems with finitely many periodic orbits and certain homological resonance relations on the actions and indices of the orbits. The ultimate goal of these projects is to better understand the underlying reasons for the abundance of periodic orbits of Hamiltonian systems. The methods to be utilized by the PI are essentially Floer-theoretic in nature. An essential new ingredient comes from the PI?s recent work, joint with Gurel, on periodic orbits of Hamiltonian diffeomorphisms with a hyperbolic fixed point.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 is among the main objectives of the proposed research. One novel ingredient of the proposed research is a new form of interaction between local and global features of Hamiltonian systems. For instance, the PI proposes to show that Hamiltonian systems with a certain type of local behavior must necessarily have infinitely many periodic orbits. The proposed research has potential applications to physics and mathematical aspects of mechanics.