Symplectic Topology of Hamiltonian Systems with Infinitely Many Periodic Orbits

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AbstractAward: DMS-0906204Principal Investigator: Basak GurelThis award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The main long-term objective of the proposed work is to examine thereason for the existence of infinitely many periodic orbits for avariety of Hamiltonian dynamical systems. The first part of theproposal is comprised of several interconnected projects addressingthis question for certain classes of Hamiltonian diffeomorphisms andalso for specific Hamiltonian systems such as twisted geodesicflows. The PI will tackle these problems by making use of methods fromsymplectic topology including Floer homological techniques, spectralinvariants, Hamiltonian Ljusternik-Schnirelman theory, the Seidelrepresentation as well as methods from differential geometry such ash-principles. Among more specific tools pertinent to thisinvestigation are local Floer homology, the mean index, the propertiesof action and index spectra and of action and index gaps, andgeometric and dynamical properties of the symplectically degeneratemaxima. The techniques utilized by the PI also have applicationsbeyond establishing the existence of infinitely many periodicorbits. In particular, the PI's approach to the projects in the secondpart of the proposal draws heavily on one of her recent works on theperiodic orbits problem. The PI outlines a method of proving that thediameter of the group of Hamiltonian diffeomorphisms is infinite formany symplectic manifolds, including some new instances. Another groupof problems considered in this proposal concerns the symplectictopology of coisotropic submanifolds and some of its applications.Hamiltonian systems constitute a broad class of physical systems wheredissipative forces can be disregarded. For example, the planetarymotion in celestial mechanics and the motion of a charged particle ina magnetic field are usually treated as Hamiltonian systems. Onegeneral, but not universal, feature of such systems is that they tendto have numerous periodic orbits. Corresponding to the cyclic motion,this is the simplest dynamical phenomenon after equilibrium and aninvestigation of periodic orbits of a system is crucial inunderstanding its global behavior. For a broad class of Hamiltoniansystems, the number of periodic orbits is known to be infinite and thisis thought to be the case for many (but not all) Hamiltoniansystems. Yet, establishing the existence of periodic orbits oftenrequires advanced and powerful mathematical tools. The proposalfocuses on the existence problem for infinitely many periodic orbits ofHamiltonian dynamical systems in a variety of settings and onapplications of the techniques used by the PI to attack this problemto some other related questions. The projects in the last part of theproposal concern a certain class of spaces which arise, for instance,in the study of Hamiltonian systems with symmetries. The proposed workis related to and has potential applications in mathematical physics,and geometric and quantum mechanics.