An infinite-dimensional version of the Poincaré-Birkhoff theorem on the Hilbert cube

We propose a version of the Poincaré-Birkhoff theorem for infinite-dimensional Hamiltonian systems, which extends a recent result by Fonda and Ureña. The twist condition, adapted to a Hilbert cube, is spread on a sequence of approximating finite-dimensional systems. Some applications are proposed to pendulum-like systems of infinitely many ODEs. We also extend to the infinite-dimensional setting a celebrated theorem by Conley and Zehnder.