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

Stefan Matijević

TL;DR

The paper proves a natural isomorphism between the filtered positive $S^1$-equivariant symplectic homology of a convex domain and the filtered singular $S^1$-equivariant homology derived from Clarke's dual functional. It deduces that the Gutt–Hutchings capacities $c_k^{GH}$ coincide with the Ekeland–Hofer spectral invariants $s_k$ for convex domains, yielding a characterization of Besse and Zoll domains via these capacities. It further shows that the barcode entropy of Clarke's dual functional bounds the topological entropy of the boundary Reeb flow, with equality in dimension four; the work also connects these invariants through a detailed framework of half-cylinder moduli spaces, Fredholm theory, and continuation maps to bridge Floer theory and Clarke duality in the convex setting.

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 $\mathbb{R}^4$.

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

TL;DR

The paper proves a natural isomorphism between the filtered positive -equivariant symplectic homology of a convex domain and the filtered singular -equivariant homology derived from Clarke's dual functional. It deduces that the Gutt–Hutchings capacities coincide with the Ekeland–Hofer spectral invariants for convex domains, yielding a characterization of Besse and Zoll domains via these capacities. It further shows that the barcode entropy of Clarke's dual functional bounds the topological entropy of the boundary Reeb flow, with equality in dimension four; the work also connects these invariants through a detailed framework of half-cylinder moduli spaces, Fredholm theory, and continuation maps to bridge Floer theory and Clarke duality in the convex setting.

Abstract

We prove that the filtered positive (-equivariant) symplectic homology of a convex domain is naturally isomorphic to the filtered singular (-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 . In particular, this barcode entropy coincides with the topological entropy for convex domains in .

Paper Structure

This paper contains 12 sections, 43 theorems, 726 equations.

Table of Contents

  1. Floer complex of a Hamiltonian
  2. Morse complex of the dual action functional
  3. Cauchy-Riemann type linear operators on negative half-cylinders
  4. Half-cylinder moduli spaces
  5. Dimension of half-cylinder moduli spaces
  6. Compactness of half-cylinder moduli space
  7. Moduli spaces of homotopy half-cylinders
  8. Chain complex isomorphism
  9. In between Clarke's duality
  10. Positive ($S^1$-equivariant) symplectic homology and autonomous Hamiltonians
  11. Positive ($S^1$-equivariant) symplectic homology and Clarke's duality
  12. Gutt-Hutchings capacities and Ekeland-Hofer spectral invariants

Key Result

Proposition 1.6

Let $H \in \mathcal{H}_F^{S^1, N}(\widetilde{f}_N, g_N)$. Let $U'$ and $U"$ be open subsets of $\mathbb{R}$ and $\mathbb{T}$, respectively, such that $J$ is a uniformly bounded $\omega_0$-compatible almost complex structure smoothly depending on $(s, t, z) \in U' \times U" \times S^{2N+1}$. If $(u,

Theorems & Definitions (100)

  • Remark
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 1.6
  • Proposition 1.7
  • proof
  • Lemma 1.8
  • ...and 90 more