A Poincar\'e-Birkhoff Theorem for Asymptotically Unitary Hamiltonian Diffeomorphisms

Abstract

We prove that an asymptotically linear Hamiltonian diffeomorphism of the standard symplectic vector space, which is non-degenerate and unitary at infinity and approaches its linear map at infinity quickly enough, has infinitely many periodic points, provided that it satisfies a natural twist condition inspired by the classical Poincar\'e-Birkhoff theorem.