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Generalizations of Ekeland-Hofer and Hofer-Zehnder symplectic capacities and applications. (English) Zbl 1520.53072

The Hofer-Zehnder symplectic capacity is a symplectic invariant that may be used to establish the existence of closed characteristics on the energy surfaces of a symplectic manifold \((M,\omega)\). In Subsection 1.1 analogs of Ekeland-Hofer symplectic capacities are constructed, while in Subsection 1.2 the construction of analogs of Hofer-Zehnder symplectic capacities based on the class of Hamiltonian boundary value problems is given. All of this is motivated by the work of F. H. Clarke [SIAM J. Control Optim. 20, 355–359 (1982; Zbl 0506.58012)] and I. Ekeland [Convexity methods in Hamiltonian mechanics. Berlin etc.: Springer-Verlag (1990; Zbl 0707.70003)]. Subsection 1.3 is devoted to application to Hamiltonian dynamics, although it does not contain a single example. Subsection 1.4 is devoted to an extension of E. Neduv’s theorem [Math. Z. 236, No. 1, 99–112 (2001; Zbl 0967.37031)], where it was shown that differentiability of the Hofer-Zehnder capacity can be used to derive some results on fixed period problems of Hamiltonian systems. Similarly differentiability also holds for the Hofer-Zehnder capacity with the boundary value problem. Sections 2–8 are used to prove the theorems stated in Section 1.

MSC:

53D35 Global theory of symplectic and contact manifolds
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57R17 Symplectic and contact topology in high or arbitrary dimension
37J12 Fixed points and periodic points of finite-dimensional Hamiltonian and Lagrangian systems
37J39 Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.)
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