Quasi-states and symplectic intersections

Abstract

We establish a link between symplectic topology and a recently emerged branch of functional analysis called the theory of quasi-states and quasi-measures (also known as topological measures). In the symplectic context quasi-states can be viewed as an algebraic way of packaging certain information contained in Floer theory, and in particular in spectral invariants of Hamiltonian diffeomorphisms introduced recently by Yong-Geun Oh. As a consequence we prove a number of new results on rigidity of intersections in symplectic manifolds. This work is a part of a joint project with Paul Biran.

Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms

“…Moreover, in this case the formula e;0 .H / D lim k!1 .eI kH / 0 =k defines a function e;0 W C.M / ! R satisfying the axioms of a symplectic quasistate, as defined in Entov and Polterovich [13]. These constructions have had many interesting applications, eg to the structure of e…”

“….eI kF / Á =k defines a symplectic quasistate in the sense of Entov and Polterovich [13]; given Proposition 3.13 and Corollary 5.5 the proof of the quasistate axioms for Á;e is an exact replication of [13,Section 6].…”

Py-calabi quasi-morphisms and quasi-states on orientable surfaces of higher genus

“…In our case, M is a closed surface of genus g ≥ 2 hence Ham(M, ω) is simply connected [11,19] and the extension splits.…”

“…In the following, M will be an oriented closed surface of genus g ≥ 2, equipped with a symplectic form ω, and µ is Py's quasimorphism given in Theorem 1.6. In [11] M. Entov and L. Polterovich construct a quasi-state from a Calabi quasi-morphism, Our goal is to show that this procedure is applicable to Py's Calabi quasi-morphism. In the following, we assume that the total area of M, denoted by V ol(M), is equal to 2g − 2.…”

“…In [11] M. Entov and L. Polterovich establish an unexpected link between a group-theoretic notion of quasi-morphism, which has been found useful in symplectic geometry, and a recently emerged branch of functional analysis called the theory of quasi-states and quasi-measures. In this paper we show this connection for a recently discovered, due to P. Py [18], Calabi quasi-morphism on orientable surfaces of higher genus.…”