Ergodic Deformations of Nonlinear Hamilton Systems and Local Homeomorphism of Metric Spaces

Abstract

The orbits of slowly perturbed Hamilton systems and the associated ergodic deformations of Lagrange manifolds are studied. The main results are based on the Mather approach [18, 19] to the construction of the homologies of invariant probabilistic measures, which minimize some Lagrange functionals, and on the elliptic Gromov–Salamon–Zehnder–Floer theory [7, 9, 12, 20, 26] of the construction of invariant manifolds. We have constructed the invariant submanifolds, which are the supports of invariant ergodic measures and have a structure of locally homeomorphic metric spaces. We analyze the problem of construction of efficient criteria of their global homeomorphism, which was posed by Professor A. M. Samoilenko during the study of ergodic deformations of nonlinear Hamilton systems and their adiabatic invariants. It is established that the mapping f : X → Y from a linearly connected Hausdorff space X onto a simply connected (in particular, contractible) space Y is a homeomorphism iff f is local and homeomorphic, and the preimage f⁻¹(y) of every point y ∈ Y is a nonempty compact subset in X.