Publication
Title
Homoclinic Floer homology via direct limits
Author
Abstract
Let (Mω) be a two dimensional symplectic manifold, φ : M → M a symplectomorphism with hyperbolic fixed point x and transversely intersecting stable and unstable manifolds Ws(φ, x) ∩ Wu(φ, x) =: H(φ, x). The intersection points are called homoclinic points, and the stable and unstable manifold are in this situation Lagrangian submanifolds. For this Lagrangian intersection problem with its infinite number of intersection points and wild oscillation behavior, we first define a Floer homology generated by finite sets of so-called contractible homoclinic points. This generalizes very significantly the Floer homologies generated by (semi)primary points defined by us in earlier works. Nevertheless these Floer homologies only consider quite ‘local’ aspects of Ws(φ, x)∩ Wu(φ, x) since their generator sets are finite, but the number of all contractible homoclinic points is infinite. To overcome this issue, we construct a direct limit of these ‘local’ homoclinic Floer homologies over suitable index sets. These direct limits thus accumulate the information gathered by the finitely generated local’ homoclinic Floer homologies.
Language
English
Publication
arXiv , 2024
DOI
10.48550/ARXIV.2402.12345
Volume/pages
27 p.
Full text (Publisher's DOI)
Full text (open access)
UAntwerpen
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Publications with a UAntwerp address
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Creation 20.02.2024
Last edited 22.02.2024
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