On the action spectrum for closed symplectically aspherical manifolds Pacific J

By M. Schwarz. 2000

348 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Computer Graphics and Computer-Aided Design, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d3929b0474e35b020d6e38113229c79fa76efd83

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Citations in the corpus

Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules (2014, background): “One readily checks that for ς(B) := ck, the corresponding value ς♯(φ) is a spectral invariant of the Hamiltonian diffeomorphism φ (see Schwarz, [43]).”
Quantum Structures for Lagrangian Submanifolds (2007, methodology): “We refer the reader to [59, 53, 51, 52, 46, 45, 44] for the foundations of the theory of spectral invariants.”
Calabi quasi-morphisms for some non-monotone symplectic manifolds (2005, methodology): “Following the works of Viterbo [28], Schwarz [26], and Oh [18], we recall the definition of a spectral invariant c which plays a central role in the proof.”
Growth of maps, distortion in groups and symplectic geometry (2001, methodology): “= 0 for all t∈ [0;1]. Let Fix 0 be the set of all ﬁxed points xof fsuch that the orbit {ftx} is contractible. Deﬁne the action spectrum of fas spectrum(f) = {A(F,x) | x∈ Fix 0} ⊂ R . It is known (see [Sch]) that the action spectrum of f does not depend on the choice of a Hamiltonian path generating ”
Gromov-Witten invariants and pseudo symplectic capacities (2001, background): “dition, it seems to be unclear whether C(0) HZ( resp. Cb (0) HZ) is diﬀerent from the Hofer-Zehnder capacity CHZ( resp. the π 1-sensitive Hofer-Zehnder capacity, denoted by C¯HZ in [Lu1] and C◦ HZ in [Sc] respectively, and the variant C′ HZ in [McSl]). But we have the following clear relations (1.12”
Hofer–Zehnder capacity and length minimizing Hamiltonian paths (2001, background): “Thus our definition corresponds to what Schwarz [27] calls the π1-sensitive Hofer–Zehnder capacity.”
Action and index spectra and periodic orbits in Hamiltonian dynamics (2008, methodology): “In fact, all symplectic topological proofs of the existence of infinitely many periodic orbits rely on the analysis of the action or index spectrum; see, eg, Ginzburg [20], Hingston [30], Hofer and Zehnder [34], Salamon and Zehnder [50], Schwarz [52] and Viterbo [56].”
Rigid subsets of symplectic manifolds (2007, background): “Recall from [59] that c([T ], Fα) > c([point], Fα).”
THE SHARP ENERGY-CAPACITY INEQUALITY (2008, methodology): “Schwarz [26] (in the symplectically aspherical case) and Oh (for general closed (M,ω)) have constructed a map ρ : C((R/Z) ×M) × (QH(M,ω) \ {0}) → R.”
LAGRANGIAN INTERSECTIONS AND THE SERRE SPECTRAL SEQUENCE (2004, methodology): “An interesting such approach is provided by Schwarz [30],[29] and is based on the pair of pants product.”
Conjugation-invariant norms on groups of geometric origin (2007, methodology): “Indeed combining Theorem 7.1 in [12] with the fact that the group Ham( T 2 ) is simply connected one gets that | µ ( φψ for all φ, ψ ∈ Ham( T 2 ), where µ is the appropriate asymptotic spectral invariant (we refer to [24, 19, 17] for preliminaries on spectral invariants).”
Persistent homology and Floer-Novikov theory (2015, methodology): “ld in rather restricted circumstances. While the ﬁltered Floer groupshave been studied since the early 1990’s and have been a signiﬁcant tool in symplectic topology since that time (see e.g. [FlH94], [Sc00], [EP03], [Oh05], [U13], [HLS15]), it is only very recently that they have been considered fro”
Calabi quasimorphism and quantum homology (2002, background): “Indeed, according to [28] and [41], c(A, 1G̃) = 0 for any singular homology class A 6= 0.”
Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds (2004, methodology): “ classical mini-max procedures. It turns out that the Floer homology theory in the chain level is the right framework for this purpose. We would like to mention that for the exact case as in [Oh1,2], [Sc] or more generally for the rational case where the period group Γω ⊂ R and so the action spectru”
Chain level Floer theory and Hofer's geometry of the Hamiltonian diffeomorphism group (2001, methodology): “The following is proved in [Oh6] (see [ Sc ] for”