9 research outputs found
A comparison of symplectic homogenization and Calabi quasi-states
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
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
Quasi-states, quasi-morphisms, and the moment map
We prove that symplectic quasi-states and quasi-morphisms on a symplectic
manifold descend under symplectic reduction on a superheavy level set of a
Hamiltonian torus action. Using a construction due to Abreu and Macarini, in
each dimension at least four we produce a closed symplectic toric manifold with
infinite dimensional spaces of symplectic quasi-states and quasi-morphisms, and
a one-parameter family of non-displaceable Lagrangian tori. By using McDuff's
method of probes, we also show how Ostrover and Tyomkin's method for finding
distinct spectral quasi-states in symplectic toric Fano manifolds can also be
used to find different superheavy toric fibers.Comment: 22 pages, 7 figures; v3: minor corrections, added remarks, and
altered numbering scheme to match published version. To appear in
International Mathematics Research Notice
Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds
We use quantum and Floer homology to construct (partial) quasi-morphisms on
the universal cover of the group of compactly supported Hamiltonian
diffeomorphisms for a certain class of non-closed strongly semi-positive
symplectic manifolds . This leads to construction of (partial)
symplectic quasi-states on the space of continuous functions on that are
constant near infinity. The work extends the results by Entov and Polterovich,
which apply in the closed case.Comment: 38 pages;v2: introduction rewritten, section 3.6 concerning open
manifolds added, several typos correcte
Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization
41 pages.International audienceFor a closed connected manifold N, we construct a family of functions on the Hamiltonian group 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 quasi-morphisms and quasi-states of Entov and Polterovich. The families are parametrized by the first real cohomology of N. In the case N=T^n the family of functions on G coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of G, to Aubry-Mather theory, to restrictions on Poisson brackets, and to symplectic rigidity