A Poincaré–Birkhoff theorem for Hamiltonian flows on nonconvex domains

Abstract

We present a higher-dimensional version of the Poincaré-Birkhoff theorem which applies to Poincaré time maps of Hamiltonian systems. The maps under consideration are neither required to be close to the identity nor to have a monotone twist. The annulus is replaced by the product of an N -dimensional torus and the interior of a (N − 1)-dimensional (not necessarily convex) embedded sphere; on the other hand, the classical boundary twist condition is replaced by an avoiding rays condition.

A topological degree theory for rotating solutions of planar systems

“…The theorem shows the existence of two fixed-points for area-preserving maps on the annulus, satisfying a twist condition of the boundary. This result is particularly suited to the study of periodic solutions of periodic Hamiltonian flows, for which a generalization in higher dimension has been recently proposed by A. Fonda and A. Ureña [16,17,13]. Yet, the area-preserving assumption is quite restrictive, albeit crucial for the existence of fixed-points.…”