Multiplicity of periodic orbits for dynamically convex contact forms
Miguel Abreu,Leonardo Macarini +1 more
TLDR
In this paper, a lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical is given.Abstract:
We give a sharp lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical. Several consequences of this result are obtained, like a new proof that every bumpy Finsler metric on $$S^n$$
carries at least two prime closed geodesics, multiplicity of elliptic and non-hyperbolic periodic orbits for dynamically convex contact forms with finitely many geometrically distinct contractible closed orbits and precise estimates of the number of even periodic orbits of perfect contact forms. We also slightly relax the hypothesis of dynamical convexity. A fundamental ingredient in our proofs is the common index jump theorem due to Y. Long and C. Zhu.read more
Citations
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Journal ArticleDOI
Lusternik–Schnirelmann theory and closed Reeb orbits
TL;DR: In this article, a variant of Lusternik-Schnirelmann theory for the shift operator in equivariant Floer and symplectic homology was developed, and it was shown that the spectral invariants are strictly decreasing under the action of shift operator when periodic orbits are isolated.
Journal ArticleDOI
Multiplicity of closed Reeb orbits on prequantization bundles
TL;DR: In this paper, the authors established multiplicity results for geometrically distinct contractible closed Reeb orbits of non-degenerate contact forms on a broad class of prequantization bundles.
Journal ArticleDOI
Non-hyperbolic closed characteristics on non-degenerate star-shaped hypersurfaces in ${\bf R}^{2n}$
TL;DR: In this article, it was shown that for every index perfect non-degenerate compact star-shaped hypersurface Σ ⊂ R2n, there exist at least n non-hyperbolic closed characteristics with even Maslov-type indices on Σ when n is even.
Posted Content
Multiplicity of Closed Reeb Orbits on Prequantization Bundles
TL;DR: In this paper, the authors established multiplicity results for geometrically distinct contractible closed Reeb orbits of non-degenerate contact forms on a broad class of prequantization bundles.
Journal ArticleDOI
Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in $$\mathbf{R}^{2n}$$ R 2 n
Huagui Duan,Hui Liu +1 more
TL;DR: In this paper, Wang et al. generalized Ekeland-Hofer theory and index iteration theory to star-shaped hypersurfaces and showed that if a compact star-shape hypersuface satisfying a suitable pinching condition carries exactly two geometrically distinct closed characteristics, then both of them must be elliptic.
References
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Book
J-Holomorphic Curves and Symplectic Topology
Dusa McDuff,Dietmar Salamon +1 more
TL;DR: The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985 as mentioned in this paper, and its applications include many key results in symplectic topology.
Book ChapterDOI
Introduction to Symplectic Field Theory
TL;DR: Symplectic Field Theory (SFT) as mentioned in this paper provides an approach to Gromov-Witten invariants of symplectic manifolds and their Lagrangian submanifolds in the spirit of topological field theory.
Journal ArticleDOI
Periodic solutions of hamiltonian systems
TL;DR: In this paper, the existence of periodic solutions of Hamiltonian systems of ordinary differential equations is proved in various settings, including free and forced vibration problems, where the period is fixed, and the proofs involve finite dimensional approximation arguments, variational methods, and appropriate estimates.
Journal ArticleDOI
The Maslov index for paths
Joel W. Robbin,Dietmar Salamon +1 more
TL;DR: In this paper, the authors define a Maslov index for any path regardless of where its endpoints lie, which is invariant under homotopy with fixed endpoints and additive for catenations.
Journal ArticleDOI
Morse theory for periodic solutions of hamiltonian systems and the maslov index
Dietmar Salamon,Eduard Zehnder +1 more
TL;DR: In this article, the authors prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over q(M).