  In this paper, we treat an open problem related to the number of periodic
orbits of Hamiltonian diffeomorphisms on closed symplectic manifolds.
Hofer-Zehnder conjecture states that a Hamiltonian diffeomorphisms has
infinitely many periodic orbits if it has "homologically unnecessary periodic
orbits"". For example, non-contractible periodic orbits are homologically
unnecessary periodic orbits because Floer homology of non-contractible periodic
orbits is trivial. We prove Hofer-Zehnder conjecture for non-contractible
periodic orbits for very wide classes of symplectic manifolds.
