  We prove a generalization of the Conley conjecture: Every Hamiltonian
diffeomorphism of a closed symplectic manifold has infinitely many periodic
orbits if the first Chern class vanishes over the second fundamental group. In
particular, we this removes the rationality condition from similar results. The
proof in the irrational case involves several new ideas including the
definition and the properties of the filtered Floer homology for Hamiltonians
on irrational manifolds. We also develop a method of localizing the filtered
Floer homology for short action intervals using a direct sum decomposition,
where one of the summands only depends on the behavior of the Hamiltonian in a
fixed open set.
