Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization

TL;DR

This paper constructs new functions on the Hamiltonian group and smooth functions of cotangent bundles, extending the concepts of quasi-morphisms and quasi-states, with applications to symplectic geometry and dynamical systems.

Contribution

It introduces a family of functions on cotangent bundles that generalize partial quasi-morphisms and quasi-states, linking to symplectic homogenization and various geometric applications.

Findings

Functions satisfy properties similar to Entov-Polterovich quasi-morphisms and quasi-states.

In the case of N=T^n, functions coincide with Viterbo's symplectic homogenization.

Applications include insights into symplectic rigidity and Aubry-Mather theory.

Abstract

For 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.