Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization

For a closed connected manifold N, we construct a family of functions on the Hamiltonian group G of the cotangent bundle T ^* N, and a family of functions on the space of smooth functions with compact support on T ^* N. These satisfy properties analogous to those of partial quasimorphisms and quasistates of Entov and Polterovich. The families are parametrized by the first real cohomology of N. In the case N=T ⁿ the family of functions on G coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of G, to Aubry-Mather theory, to restrictions on Poisson brackets, and to symplectic rigidity.