Local Symplectic Homology of Reeb Orbits

TL;DR

This paper establishes isomorphisms in local symplectic homology for simple isolated Reeb orbits, linking local Floer homology to return maps and clarifying the concept of symplectically degenerate maxima.

Contribution

It proves two isomorphisms in local symplectic homology for simple Reeb orbits, connecting Floer homology and return maps, and clarifies definitions of symplectically degenerate maxima.

Findings

Isomorphism between local $S^1$-equivariant symplectic homology and local Hamiltonian Floer homology.

Equivalence of different definitions of symplectically degenerate maximum.

Relation of local Floer homology to the return map of Reeb orbits.

Abstract

In this paper we prove two isomorphisms in the local symplectic homology of a simple, which is to say 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 isomorphism we prove in $S^1$-equivariant symplectic homology can be stated succinctly as the local $S^1$-equivariant symplectic homology of a simple isolated Reeb orbit is isomorphic to the local Hamiltonian Floer homology of the return map. We also prove the equivalence of two different definitions of a Reeb orbit being a symplectically degenerate maximum.