Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization

Abstract

For a closed connected manifold $N$, we construct a family offunctions on the Hamiltonian group $\mathcal{G}$ of the cotangent bundle$T^*N$, and a family of functions on the space of smooth functionswith compact support on $T^*N$. These satisfy properties analogous tothose of partial quasimorphisms and quasistates of Entov andPolterovich. The families are parametrized by the first realcohomology of $N$. In the case $N=\mathbb{T}^n$ the family of functions on$\mathcal{G}$ coincides with Viterbo's symplectic homogenizationoperator. These functions have applications to the algebraic andgeometric structure of $\mathcal{G}$, to Aubry--Mather theory, to restrictionson Poisson brackets, and to symplectic rigidity.