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

We study Floer homology groups for asymptotically linear Hamiltonian diffeomorphisms on standard symplectic space. Our main application is the proof of a theorem resembling the classical Poincar\'e-Birkhoff theorem: we show that if an asymptotically linear Hamiltonian diffeomorphism, non-degenerate and unitary at infinity, admits a homologically visible fixed point whose mean Conley-Zehnder index is different from the mean Conley-Zehnder index at infinity, then it has infinitely many periodic orbits.