  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 quasi-morphisms and quasi-states 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.
