Periodic orbits of Hamiltonian systems and symplectic topology of coisotropic submanifolds

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

Abstract Award: DMS-1007149 Principal Investigator: Viktor Ginzburg The 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.