UC Santa Cruz

The main theme of this thesis is the interaction between symplectic topology and Hamiltonian and symplectic dynamics.

The first problem considered in this thesis concerns symplectic topology of coisotropic submanifolds. We revisit the definition of the coisotropic Maslov index and prove a Maslov index rigidity result for stable coisotropic submanifolds in a broad class of ambient symplectic manifolds. Furthermore, we establish a nearby existence theorem for the same class of ambient manifolds. The main tools used to achieve these goals are Hamiltonian Floer homology and Kerman's pinned action selector.

The existence of periodic orbits of symplectomorphisms lies at the center of the second problem we consider. We are interested in a variant of the Conley conjecture which asserts the existence of infinitely many periodic orbits of a symplectomorphism if it has a fixed point which is unnecessary in some sense. More specifically, we show that, for a certain class of closed monotone symplectic manifolds, any symplectomorphism isotopic to the identity with a hyperbolic fixed point must necessarily have infinitely many periodic orbits as long as the symplectomorphism satisfies some constraints on the flux. The main tool used to prove this result is Floer homology for symplectomorphisms, i.e. the Floer-Novikov homology.