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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.

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