Journal of Modern Dynamics (JMD)

Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization

Pages: 205 - 249, Issue 2, April 2012      doi:10.3934/jmd.2012.6.205

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Alexandra Monzner - Fakultät für Mathematik, TU Dortmund, Dortmund, Germany (email)
Nicolas Vichery - CMLS École Polytechnique, Palaiseau, France (email)
Frol Zapolsky - Mathematisches Institut der Ludwig-Maximilian-Universität, Munich, Germany (email)

Abstract: 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.

Mathematics Subject Classification:  Primary: 53D40, 37J50.

Received: January 2012;      Available Online: August 2012.