Finite energy foliations, Floer and contact homology in low dimensions, Hamiltonian dynamics, braids
Final Report Abstract
A Floer theory supported by binding orbits and the construction of finite energy foliations (Subproject A). A main objective of this project was to develop a Floer theory for Hamiltonian surface diffeomorphisms where the generators of the chain complex are a collection of periodic orbits that have features that make them good candidates for the binding orbits of a finite energy foliation (of the symplectization of a mapping torus). Another aim was to use such a frame-work to construct finite energy foliations. We obtained some partial results in this direction, but there remains to show invariance of the resulting Floer homology groups. Together with J. Fish, Siefring constructed finite energy foliations using other methods, with potential applications to the restricted three-body problem. An unanticipated difficulty that arose in proving invariance of the Floer homology groups, due to working with exponential weights, lead to technical questions about the asymptotic behaviour of finite energy pseudo-holomorphic curves near punctures. In Hojo’s Masters-Thesis a refinement of the asymptotic formula is proven under mild simplifying assumptions. On the other hand, Siefring showed that the asymptotic behaviour of a pseudo-holomorphic curve at a puncture may include non-uniqueness of the asymptotic limiting periodic orbits, even if the global energy is finite. A Floer theory transverse to the binding orbits of a finite energy foliation (Subproject B). Another main objective in this project was to develop a Floer theory for Hamiltonian surface diffeomorphisms where the generators of the chain complex are periodic orbits transverse to the leaves of an auxiliary finite energy foliation. The core new ideas for mapping tori of closed surfaces of positive genus are well developed in the Ph.D.-Thesis of J. Ojeda: He establishes a chain complex, proves an invariance theorem for the resulting “transverse Floer homology groups”, and will compute them and use this to give an application to surface dynamics which appears to be an interesting generalisation of a fundamental result of J. Franks.
Publications
- Holomorphic curves in the presences of holomorphic hypersurface foliations
A. Moreno and R. Siefring
- Finite-energy pseudoholomorphic planes with multiple asymptotic limits, Math. Ann. 368 (2017), no. 1-2, 367–390
R. Siefring
(See online at https://doi.org/10.1007/s00208-016-1478-y) - Connected sums and finite energy foliations I: Contact connected sums, J. Symplectic Geom. 16 (2018), no. 6, 1639–1748
J. Fish and R. Siefring
(See online at https://doi.org/10.4310/JSG.2018.v16.n6.a4) - Transversal homology for foliated Hamiltonian systems, Ph.D. Thesis, Ruhr University Bochum (2021). 165 S.
J. S. Ojeda
(See online at https://doi.org/10.13154/294-8344)
