# American Institute of Mathematical Sciences

April  2012, 6(2): 205-249. doi: 10.3934/jmd.2012.6.205

## Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization

 1 Fakultät für Mathematik, TU Dortmund, Dortmund, Germany 2 CMLS École Polytechnique, Palaiseau, France 3 Mathematisches Institut der Ludwig-Maximilian-Universität, Munich, Germany

Received  January 2012 Published  August 2012

For a closed connected manifold $N$, we construct a family of functions on the Hamiltonian group $\mathcal{G}$ 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 $N=\mathbb{T}^n$ the family of functions on $\mathcal{G}$ coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of $\mathcal{G}$, to Aubry--Mather theory, to restrictions on Poisson brackets, and to symplectic rigidity.
Citation: Alexandra Monzner, Nicolas Vichery, Frol Zapolsky. Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization. Journal of Modern Dynamics, 2012, 6 (2) : 205-249. doi: 10.3934/jmd.2012.6.205
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