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

TL;DR

This paper establishes a deep connection between symplectic homology and Clarke's dual functional for convex domains, leading to new insights into symplectic capacities, Reeb flow dynamics, and entropy bounds.

Contribution

It proves an isomorphism between positive $S^1$-equivariant symplectic homology and Clarke's dual homology, linking capacities and flow dynamics in convex domains.

Findings

Gutt-Hutchings capacities match Ekeland-Hofer spectral invariants.

Besse convex domains are characterized by their capacities.

Barcode entropy provides a lower bound for Reeb flow entropy.

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 $\mathbb{R}^4$.