Positive (S¹-equivariant) symplectic homology of convex domains, higher capacities, and Clarke's duality

2024/10/17 by Matijević, Stefan · 3 citations
#Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2410.13673

Abstract

We prove that the filtered positive (S¹-equivariant) symplectic homology of a convex domain is naturally isomorphic to the filtered singular (S¹-equivariant) homology induced by Clarke's dual functional associated with the convex domain. As a result, we prove that the Gutt-Hutchings capacities coincide with the spectral invariants introduced by Ekeland-Hofer for convex domains. From this identification, it follows that Besse convex domains can be characterized by their Gutt-Hutchings capacities, which implies that the interiors of Besse-type convex domains encode information about the Reeb flow on their boundaries. Moreover, as a corollary of the aforementioned isomorphism, we deduce that the barcode entropy associated with the singular homology induced by Clarke's dual functional provides a lower bound for the topological entropy of the Reeb flow on the boundary of a convex domain in ℝ²ⁿ. In particular, this barcode entropy coincides with the topological entropy for convex domains in ℝ⁴.

Cited by

• A note on the wild symplectic ellipsoids
• Periodicity characterization by capacities for star-shaped domains
• Hamiltonian linking and Symplectic packing

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