Positive ($S^1$-equivariant) symplectic homology of convex domains, higher capacities, and Clarke's duality

Abstract

We prove that the filtered positive ($S^1$-equivariant) symplectic homology of a convex domain is naturally isomorphic to the filtered singular ($S^1$-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 $\mathbb{R}^{2n}$. In particular, this barcode entropy coincides with the topological entropy for convex domains in $\mathbb{R}^4$.