Generic Reeb flows on 3-manifolds have homoclinic connections on every hyperbolic orbit, can be given embedded Birkhoff sections with prescribed boundary and Legendrian content, and every Legendrian knot has infinitely many Reeb chords.
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Homoclinic orbits, Reeb chords and nice Birkhoff sections for Reeb flows in 3D
Generic Reeb flows on 3-manifolds have homoclinic connections on every hyperbolic orbit, can be given embedded Birkhoff sections with prescribed boundary and Legendrian content, and every Legendrian knot has infinitely many Reeb chords.