Local Symplectic Homology of Reeb Orbits

2020/10/03 by Elijah Fender, Fender, Elijah · 2 citations
Mathematics · #Advanced Operator Algebra Research #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.SG #msc:53D40

paper · pdf · doi:10.48550/arxiv.2010.01438

41 pages

arxiv created 2020/10/03 · arxiv updated 2020/10/06

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 S¹-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 S¹-equivariant symplectic homology can be stated succinctly as the local S¹-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.

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• Two irrationally elliptic closed orbits of Reeb flows on the boundary of star-shaped domain in ℝ²ⁿ
• On the growth rate of Reeb orbit on star-shaped hypersurfaces

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