Homoclinic Floer homology via direct limits

TL;DR

This paper develops a new Floer homology framework for analyzing homoclinic points of symplectomorphisms on surfaces, using direct limits of local homoclinic Floer homologies to capture global dynamical information.

Contribution

It introduces a Floer homology constructed via direct limits of finite sets of contractible homoclinic points, extending previous local analyses to a more comprehensive global perspective.

Findings

Defined Floer homology from finite homoclinic point sets

Constructed direct limits to aggregate local Floer homologies

Enhanced understanding of homoclinic intersection dynamics

Abstract

Assume $M$ to be $\mathbb R^2$ or a closed surface of genus $g \geq 1$ and $\omega$ 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.