Homoclinic orbits, Reeb chords and nice Birkhoff sections for Reeb flows in 3D

TL;DR

This paper proves generic properties of Reeb flows on 3-manifolds, including the existence of homoclinic connections, global surfaces of section containing specified orbits and links, and infinitely many Reeb chords for Legendrian knots.

Contribution

It establishes generic existence of homoclinic orbits, constructs Birkhoff sections containing given orbits and links, and shows that Legendrian knots have infinitely many Reeb chords under broad conditions.

Findings

Generic Reeb flows have transverse homoclinic connections.

Existence of global surfaces of section containing specified orbits and Legendrian links.

Legendrian knots possess infinitely many Reeb chords in generic settings.

Abstract

We prove that for a $C^\infty$-generic contact form defining a given co-oriented contact structure on a closed $3$-manifold, every hyperbolic periodic Reeb orbit admits a transverse homoclinic connection in each of the branches of its stable and unstable manifolds. We exploit this result to prove that for a $C^\infty$-generic contact form defining a given co-oriented contact structure, given any finite collection $\Gamma$ of periodic Reeb orbits and any Legendrian link $L$, there exists a global surface of section (embedded Birkhoff section) for the Reeb flow that contains $\Gamma$ in its boundary, and that contains in its interior a Legendrian link that is Legendrian isotopic to $L$ by a $C^0$-small isotopy. Finally we prove that if the Reeb vector field admits a $\partial$-strong Birkhoff section then every Legendrian knot has infinitely many geometrically distinct Reeb chords, except possibly when the ambient manifold is a lens space or the sphere and the Reeb flow has exactly two periodic orbits. In particular, $C^\infty$-generically on the contact form there are infinitely many geometrically distinct Reeb chords for every Legendrian knot. In the case of geodesic flows, every Legendrian knot has infinitely many disjoint chords, without any further assumptions.