For a nice contact form, some Reeb orbit intersects a given symplectic surface with frequency at least the surface area divided by the contact volume, without any genericity assumption.
Hutchings, Elementary spectral invariants and quantitative closing l emmas for contact three-manifolds , J
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Reeb orbits frequently intersecting a symplectic surface
For a nice contact form, some Reeb orbit intersects a given symplectic surface with frequency at least the surface area divided by the contact volume, without any genericity assumption.