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A comparison of symplectic homogenization and Calabi quasi-states
Alexandra Monzner and Frol Zapolsky
We compare two functionals defined on the space of continuous functions with compact support in an open neighborhood of the zero section of the cotangent bundle of a torus. One comes from Viterbo’s symplectic homogenization, the other from the Calabi quasi-states due to Entov and Polterovich. In dimension 2 we are able to say when these two functionals are equal. A partial result in higher dimensions is presented. We also indicate a link to asymptotic Hofer geometry on T*S^1. Proofs are based on the theory of quasi-integrals and topological measures on locally compact spaces.