A comparison of symplectic homogenization and Calabi quasi-states

TL;DR

This paper compares symplectic homogenization and Calabi quasi-states functionals on cotangent bundles, establishing conditions for their equality in dimension 2 and exploring higher-dimensional cases with links to Hofer geometry.

Contribution

It provides a detailed comparison of two symplectic functionals, identifying when they coincide and extending the analysis to higher dimensions with connections to Hofer geometry.

Findings

Equality of functionals in dimension 2 under certain conditions

Partial results for higher dimensions

Connection to asymptotic Hofer geometry

Abstract

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 while 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 give 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.