Periodic orbits of Hamiltonian systems

该提案的重点是哈密顿系统的周期轨道的存在性问题和其他几个问题密切相关的PI的以前的工作。 第一组问题考虑的建议有关的推广康利猜想。这个猜想断言存在无穷多个周期点的一个辛非球面，闭流形的哈密顿同态。Conley猜想是由Hingston（对于环面）和PI建立的，并最终推广到所有具有零Chern类的闭辛流形和负单调辛流形。PI提出了各种进一步推广这些结果，以及一个程序来调查的情况下，猜想是已知的不举行。特别是，PI将研究存在无穷多个周期轨道的哈密顿同构与双曲不动点，高维广义的弗兰克斯？定理，以及Conley猜想的流形与大的最小陈数。该建议的第二部分涉及一个类似的循环问题的周期轨道的Reeb流。最后，该建议的最后一部分的主要课题是研究具有多个周期轨道的Hamilton系统，以及轨道的作用和指数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根本原因。PI使用的方法本质上是Floer理论。一个重要的新成分来自PI？Hamilton动力系统描述了许多类耗散力可以忽略不计的物理过程。例如，天体力学中的行星运动和一些电动或磁动力学过程可以，而且通常是，被视为哈密顿动力系统。位于哈密顿动力系统和辛几何的现代理论的核心的经典主题之一是周期轨道的研究（即，循环运动）。周期轨道是普遍存在的：绝大多数哈密顿系统具有周期轨道，并且对于大类系统，不同周期轨道的数目是无限的。对这一现象的分析是拟议研究的主要目标之一。 一个新的成分，拟议的研究是一种新的形式之间的相互作用的局部和全局特征的哈密顿系统。例如，PI提出证明具有某种局部行为的哈密顿系统必然具有无穷多个周期轨道。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 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.