Homoclinic Floer homology via direct limits

TL;DR

This work develops a direct-limit framework to define Floer-type homology for homoclinic points in two-dimensional symplectic dynamics. It starts with local Floer complexes on finite sets of contractible homoclinic points, equipped with a Maslov grading and a combinatorial boundary, and then cures non-closure issues via a pruning procedure, producing ∂-complete sets. By organizing these local homologies into directed systems with chain-compatible inclusions, it forms direct limits that capture global information about the homoclinic tangle. The approach provides a principled way to aggregate local dynamical data into a robust homology theory, connecting more traditional (semi)primary constructions to a broader, scalable global invariant.

Abstract

Assume $M$ to be $\mathbb R^2$ or a closed surface of genus $g \geq 1$ and $ω$ a symplectic form on $M$. Let $\varphi: M \to M$ be a symplectomorphism with hyperbolic fixed point $x$ and transversely intersecting stable and unstable manifolds $W^s(\varphi, x)$ and $ W^u(\varphi, x)$. The intersection points $W^s(\varphi, x) \cap\ W^u(\varphi, x)=:{\mathcal H}(\varphi, x)$ are called homoclinic points, and the (un)stable manifolds of a symplectomorphism are Lagrangian submanifolds. For this Lagrangian intersection problem with its wildly oscillating Lagrangian manifolds and infinite number of intersection points, we introduced in earlier works Floer homologies generated by so-called (semi)primary homoclinic points and analysed their dynamical and geometric properties. In this paper, we significantly generalise these earlier results by first defining a Floer homology generated by finite sets of contractible homoclinic points. These Floer homologies nevertheless still consider rather `local' aspects of $W^s(\varphi, x) \cap\ W^u(\varphi, x)$ since their generator sets are finite (but the number of contractible homoclinic points is infinite). To overcome this issue, we construct a direct limit of these `local' homoclinic Floer homologies over suitable index sets. These direct limits accumulate the information gathered by the finitely generated `local' homoclinic Floer homologies.

Figures (6)

• Figure 1: The intersection behaviour ('homoclinic tangle') of transversely intersecting stable (= continuous line) and unstable (= dotted line) manifold of a hyperbolic fixed point (= bold black dot).
• Figure 2: Di-gon and hearts.
• Figure 3: Geometric intuition for gluing in (a) and cutting with the two placements of the concave and convex vertices in (b). The unstable manifold $W^u$ is drawn with a dotted line and the stable manifold $W^s$ with a full line.
• Figure 4: The points $q_a$ and $q_b$ may appear as cutting partners for two different tuples $(p,r)$ and $(p,r')$ that satisfy both the requirements of Theorem \ref{['cut']}.
• Figure 5: The unstable manifold is drawn with a dotted line and the stable manifold with a continuous line. Given a homoclinic point $p$ as chosen in the figure, then we find $\lvert \{q \in \mathcal{H} \mid n(p,q) \neq 0\} \rvert = \infty$ since $n(p,q_i) \neq 0$ for at least $q_1, q_2, q_3, q_4, \dots$

Theorems & Definitions (57)

• Theorem 1.1
• Proposition 1.2
• Proposition 1.3
• Theorem 1.4
• Theorem 2.1: Hohloch hohloch1, Theorem 14, 'Gluing'
• Theorem 2.2: Hohloch hohloch1, Theorem 15, 'Cutting'
• Remark 2.3
• Lemma 2.4: Hohloch hohloch1, Lemma 21
• Example 3.1
• Example 3.2