Positive ($S^1$-equivariant) symplectic homology of convex domains, higher capacities, and Clarke's duality
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913687385669632 |
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| author | Matijević, Stefan |
| author_facet | Matijević, Stefan |
| contents | 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 $\mathbb{R}^4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_13673 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positive ($S^1$-equivariant) symplectic homology of convex domains, higher capacities, and Clarke's duality Matijević, Stefan Symplectic Geometry Differential Geometry Dynamical Systems 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 $\mathbb{R}^4$. |
| title | Positive ($S^1$-equivariant) symplectic homology of convex domains, higher capacities, and Clarke's duality |
| topic | Symplectic Geometry Differential Geometry Dynamical Systems |
| url | https://arxiv.org/abs/2410.13673 |