  The aim of this paper is to use the methods and results of symplectic
homogenization (see [V4]) to prove existence of periodic orbits and invariant
measures with rotation number depending on the differential of the Homogenized
Hamiltonian. We also prove the Conley conjecture on the cotangent bundle of the
torus. Both proofs rely on Symplectic Homogenization and a refinement of it.
