Symplectic method for the construction of ergodic measures on invariant submanifolds of nonautonomous hamiltonian systems: Lagrangian manifolds, their structure, and mather homologies

Abstract

We develop a new approach to the study of properties of ergodic measures for nonautonomous periodic Hamiltonian flows on symplectic manifolds, which are used in many problems of mechanics and mathematical physics. Using Mather’s results on homologies of invariant probability measures that minimize some Lagrangian functionals and the symplectic theory developed by Floer and others for the investigation of symplectic actions and transversal intersections of Lagrangian manifolds, we propose an analog of a Mather-type β-function for the study of ergodic measures associated with nonautonomous Hamiltonian systems on weakly exact symplectic manifolds. Within the framework of the Gromov-Salamon-Zehnder elliptic methods in symplectic geometry, we establish some results on stable and unstable manifolds for hyperbolic invariant sets, which are used in the theory of adiabatic invariants of slowly perturbed integrable Hamiltonian systems.