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

TL;DR

This work establishes that $C^ ablafty$-generic Reeb flows on closed 3-manifolds exhibit robust chaotic features: every hyperbolic periodic orbit has transverse homoclinic connections in both stable and unstable branches, enabling transfer of surface-dynamics methods to Reeb dynamics. Leveraging this, the authors construct embedded Birkhoff sections that can be constrained to include prescribed periodic orbits on the boundary and Legendrian links in the interior, using Fried’s pair-of-pants techniques and Zehnder-type conditions. They then show that, for Reeb flows with a $ abla$-strong Birkhoff section, every Legendrian knot carries infinitely many Reeb chords (with a sharp finite/infinite dichotomy in the exceptional two-orbit lens-space/sphere case), and obtain analogous results for geodesic flows where disjoint chords proliferate. The results provide a unified framework to transfer two classical lines of thought from surface dynamics to Reeb flows, with strong implications for chord conjectures, open book/open-structure analogues, and the global behavior of Reeb chord growth.

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 $Γ$ 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 $Γ$ 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.

Figures (4)

• Figure 1: The Fried sum of $P$ and $S$ that eliminates an intersection point in $L\cap S$. Here we have moved it to $z \in L'\cap R_2$. The possible nearby intersection point $z'$ of $L'$ and $R_1$ can be eliminated by sliding $L'$ along the characteristic foliation of $R_1$, after first pushing $S$ up slightly along $R_1$ to avoid creating extra intersections with $S$ when sliding $L'$.
• Figure 2: The pair of pants $P$ (before smoothing) near $Q_0$. The Reeb vector field is vertical and the contact structure approximately horizontal. We push $L_1$ away from $P$ along its characteristic foliation $\xi P$ in counterclocwise manner to obtain $L_2$ (in blue), without creating new chords. The Reeb chord intercepted by $Q_0$ is in red.
• Figure 3: The dynamics of the first-return map along $\phi^t$ on a transverse disc $D$ around a hyperbolic periodic point $z_0$.
• Figure 4: Construction of the pair of pants $P$ from the three periodic points $w_{1,1}, w_{1,2}, w_{2,2}$ in $D$.

Theorems & Definitions (15)

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