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Proof of Hofer-Wysocki-Zehnder's two or infinity conjecture
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abstract
We prove that every Reeb flow on a closed connected three-manifold has either two or infinitely many simple periodic orbits, assuming that the associated contact structure has torsion first Chern class. As a special case, we prove a conjecture of Hofer-Wysocki-Zehnder published in 2003 asserting that a smooth and autonomous Hamiltonian flow on $\mathbb{R}^4$ has either two or infinitely many simple periodic orbits on any regular compact connected energy level that is transverse to the radial vector field. Other corollaries settle some old problems about Finsler metrics: we show that every Finsler metric on $S^2$ has either two or infinitely many prime closed geodesics; and we show that a Finsler metric on $S^2$ with at least one closed geodesic that is not irrationally elliptic must have infinitely many prime closed geodesics. The novelty of our work is that we do not make any nondegeneracy hypotheses.
Forward citations
Cited by 4 Pith papers
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On the contact type conjecture for exact magnetic systems
Constructs infinite-dimensional spaces of exact magnetic systems of strong geodesic type on closed manifolds, proving existence of null-homologous embedded periodic orbits with negative action below the strict Mañé cr...
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Multiplicity of closed Reeb orbits on contact manifolds with periodic equivariant symplectic homology
On contact manifolds with periodic positive equivariant symplectic homology, nondegenerate forms satisfying weak index conditions have at least r_M simple closed Reeb orbits, with equality if and only if the form is lacunary.
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Multiplicity of closed Reeb orbits on contact manifolds with periodic equivariant symplectic homology
Sharp lower bound r_M on simple closed Reeb orbits is attained iff the form is lacunary and equals dim H_*(M/S^1; Q) for prequantizations, making r_M a contact invariant from positive equivariant symplectic homology.
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Elementary spectral invariants and three-dimensional Reeb dynamics
Elementary spectral invariants simplify embedded contact homology spectral invariants for contact three-manifolds and can be used to prove some results on periodic orbits of Reeb vector fields, with the remaining resu...
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