A quasi-integral on a locally compact space is a certain kind of (not necessarily linear) functional on the space of continuous functions with compact support of that space. We compare two quasi-integrals on an open neighborhood of the zero section of the cotangent bundle of a circle. One comes from Viterbo's symplectic homogenization, the other from the Calabi quasi-state due to Entov and Polterovich. We provide an axiomatic description of the two functionals and a necessary and sufficient condition for them to equal. 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. Finally, we give an elementary proof that a quasi-integral on a surface is symplectic.
Bibliographical noteFunding Information:
We thank Leonid Polterovich for suggesting the topic of this paper and for his interest in it, as well as for pointing out the link to asymptotic Hofer geometry, and Karl Friedrich Siburg for useful discussions and suggestions. The second author would like to thank Judy Kupferman and Marco Mazzucchelli for listening to a preliminary version of the results and for helpful suggestions. Finally, we would like to thank the anonymous referee for numerous remarks which allowed us to improve the readability of the paper. This work started during the stay of the first author at the University of Chicago, which was supported by the Martin-Schmeißer-Foundation. The first author is partially supported by the German National Academic Foundation. This work is partially supported by the NSF-grant DMS 1006610.
- Hofer geometry
- Symplectic homogenization
ASJC Scopus subject areas
- Geometry and Topology