UC Santa Cruz

In this thesis I prove two isomorphisms in the local symplectic homology of a non iterated, isolated Reeb orbit. The isomorphisms are in $S^1$-equivariant and nonequivariant symplectic homology relating the local Floer homology group of the orbit to that of the return map. The $S^1$-equivariant symplectic homology isomorphism can be stated succinctly as the local $S^1$-equivariant symplectic homology of an uniterated isolated Reeb orbit is isomorphic to the local Hamiltonian Floer homology of the return map. An application of this result gives the equivalence of two different definitions of a Reeb orbit being a symplectically degenerate maximum.