Homoclinic Floer homology via direct limits

Assume $M$ to be ${\mathbb{R}}^2$ or a closed surface of genus $g\ge 1$ and $\omega$ a symplectic form on $M$. Let $\phi :M\to M$ be a symplectomorphism with hyperbolic fixed point $x$ and transversely intersecting stable and unstable manifolds $W^s(\phi ,x)$ and $W^u(\phi ,x)$. The intersection points $W^s(\phi ,x)\cap W^u(\phi ,x)=:\mathcal{H}(\phi ,x)$ 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.