  In this paper we show how the rich algebraic formalism of
Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define
higher algebraic structures in Hamiltonian Floer theory. Using the SFT of
Hamiltonian mapping tori we show how to define a homotopy extension of the
well-known Lie bracket and discuss how it can be used to prove the existence of
multiple closed Reeb orbits. Furthermore we show how to define the analogue of
rational Gromov-Witten theory in the Hamiltonian Floer theory of open
symplectic manifolds. More precisely, we introduce a so-called cohomology
F-manifold structure in Hamiltonian Floer theory and prove that it generalizes
the well-known Frobenius manifold structure in rational Gromov-Witten theory.
