On functoriality in Floer homology

Floer theory associates to a symplectic manifold $(M,\omega)$ and a Hamilton function $H:S1\times M\rightarrow \mathbb{R}$ the symplectic invariant $\mathrm{HF}_*(H)$, namely Floer homology groups. We address the question how to assign to a symplectic map $f:(M,\omega)\rightarrow(N,\sigma)$ an induced homomorphism on Floer homology, and propose a construction for a large class of such maps. Inspection of special embeddings leads to new results on the topology and geometry of Lagrangian submanifolds.