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.
