Propagation in Hamiltonian dynamics and relative symplectic homology

By P. Biran, L. Polterovich, Dietmar Salamon. 2001

113 citations recorded by the source corpus.

The main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating Hamiltonian is sufficiently large over the zero section. The proof is based on Floer homology and on the notion of a relative symplectic capacity. Applications include results about propagation properties of sequential Hamiltonian systems, periodic orbits on hypersurfaces, Hamiltonian circle actions, and smooth Lagrangian skeletons in Stein manifolds.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/94150d77747cdca3892a2f1d64cc8bf958f0e1b0

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Citations in the corpus

Quantitative symplectic geometry (2005, unknown): “Since then, lots of new capacities have been defined [16, 30, 32, 44, 49, 59, 60, 90, 99] and they were further studied in [1, 2, 8, 9, 17, 26, 21, 28, 31, 35, 37, 38, 41, 42, 43, 46, 48, 50, 52, 56, 57, 58, 61, 62, 63, 64, 65, 66, 68, 74, 75, 76, 88, 89, 91, 92, 94, 97, 98].”
A FLOER HOMOLOGY FOR EXACT CONTACT EMBEDDINGS (2007, background): “(In fact, the construction in [Poźniak 1999] is for Lagrangian intersections; see [Biran et al. 2003] for the transfer to the periodic orbit case).”
The Künneth formula in Floer homology for manifolds with restricted contact type boundary (2004, background): “M. Their deﬁnition is closely related to the Symplectic homology of Floer, Hofer, Cieliebak and Wysocki [FH, CFH, FHW, CFHW], as well as to more recent constructions of Biran, Polterovich and Salamon [BPS] on cotangent bundles or Frauenfelder and Schlenk [FrSc] on general convex manifolds. The group”
Symplectic homology and periodic orbits near symplectic submanifolds (2002, unknown): “y thickening of the boundary of such a neighborhood. Symplectic homology was originally introduced and studied in the series of papers [CFH, CFHW, FH, FHW]. Here we follow more recent treatments from [BPS] and, more indirectly, from [Vi]. In particular, we consider the (direct limit) symplectic homo”
Relative Hofer-Zehnder capacity and periodic orbits in twisted cotangent bundles (2003, methodology): “han the one employed in this paper (see Remark 5.3). Then, assuming that this condition is superﬂuous, one can interpret Theorem 3.1 as a calculation of the relative symplectic capacity introduced in [BPS] for the class of contractible orbits. When Mis a Lagrangian submanifold and the orbits are sou”
Action and index spectra and periodic orbits in Hamiltonian dynamics (2008, background): “The idea of the proof is the squeezing trick [4; 26; 35]: the Hamiltonian H is bounded from above and below by functions H ̇ such that the monotone homotopy map for the iterated Hamiltonians H .”
Spectral invariants in Lagrangian Floer theory (2006, unknown): “We provide a way to distinguish them one from another and estimate their difference in terms of a geometric quantity.”
Erratum to “Geometry of contact transformations and domains: orderability versus squeezing” (2005, methodology): “First of all, let us recall the standard definition of the Floer homology of F (see eg [3]).”
Spectral invariants for monotone Lagrangians (2015, result): “proved in [MVZ12] based on the trick due to Abbondandolo–Schwarz [AS10]. Theorem 5 was proved by Leclercq [Lec08]for weakly exact M. The canonical isomorphism QH∗(∆) = QH∗(M) appears, for example, in [BPS03]. Various versions of Proposition 8 have appeared in the literature, see for example [EP09], ”