On functoriality in Floer homology

Floer theory associates to a symplectic manifold $(M,\omega )$ and a Hamilton function $H:S1\times M\to \mathbb{R}$ the symplectic invariant ${\mathrm{HF}}_{\ast }(H)$, namely Floer homology groups. We address the question how to assign to a symplectic map $f:(M,\omega )\to (N,\sigma )$ 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.