  We propose a twist condition, inspired by the classical Poincar\'e-Birkhoff
theorem on area-preserving maps of the annulus, which applies to asymptotically
linear Hamiltonian diffeomorphisms of linear phase space. When an
asymptotically linear Hamiltonian diffeomorphism, non-degenerate and unitary at
infinity, satisfies this twist condition, we show that it has infinitely many
periodic points.
