Local Floer Homology and the Action Gap
By V. Ginzburg, Basak Z. Gurel. 2007
82 citations recorded by the source corpus.
In this paper, we study the behavior of the local Floer homology of an isolated fixed point and the growth of the action gap under iterations. To be more specific, we prove that an isolated fixed point of a Hamiltonian diffeomorphism remains isolated for a certain class of iterations (the so-called admissible iterations) and that the local Floer homology groups for all such iterations are isomorphic to each other up to a shift of degree. Furthermore, we study the pair-of-pants product in local Floer homology, and characterize a particular class of isolated fixed points (the symplectically degenerate maxima), which plays an important role in the proof of the Conley conjecture. The proofs of these facts rely on an observation that for a general diffeomorphism, not necessarily Hamiltonian, an isolated fixed point remains isolated under all admissible iterations. Finally, we apply these results to show that for a quasi-arithmetic sequence of admissible iterations of a Hamiltonian diffeomorphism with isolated fixed points the minimal action gap is bounded from above when the ambient manifold is closed and symplectically aspherical. This theorem is a generalization of the Conley conjecture.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/93ce806a694a32849663c107941b4790d2153076
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Discrete Mathematics and Combinatorics
In Geometry and Topology
- Basak Z. Gurel
- V. Ginzburg
- Sheila Sandon
- L. Polterovich
- M. Crainic
- Rui Loja Fernandes
- Y. Eliashberg
- Michael Usher
In Mathematical Physics
- V. Ginzburg
- Basak Z. Gurel
- Sheila Sandon
- M. Crainic
- Rui Loja Fernandes
- Y. Eliashberg
- L. Polterovich
- Sang Seon Kim
Citations in the corpus
- The Conley Conjecture (2006, background): “of Ki and Hhave equal actions and, in particular, all Hamiltonians Ki have the same action spectrum and action filtration; see Section 2.3. Symplectically degenerate maxima are further investigated in [GG2]. In particular, it is shown there that condition (K3) is superfluous; see [GG2, Remark 5.5]. Th”
- Action and index spectra and periodic orbits in Hamiltonian dynamics (2008, background): “pendent of each other. In Section 3, we establish Theorems 1.1, 1.3, and 1.18, along with some auxiliary results on the action and index spectra. Theorem 1.4 is proved in Section 4, following closely [GG3]. In Section 5, we prove Theorem 1.17, which the previous results rely on in the case where the”
- Lusternik–Schnirelmann theory and closed Reeb orbits (2016, background): “ices are quasimorphisms Sp(2f m) → Z. These indices are of particular interest to us because they bound the support of the local Floer homology of an isolated periodic orbit; cf. Proposition 2.20 and [GG10]. 4.1.2. Basic properties. Let us now list the properties of the Conley–Zehnder type indices, ”