Large cycles in hamiltonian systems

The existence of large cyclic flows in the phase space of systems which are governed by a Hamiltonian under zero-temperature parallel dynamics is proven. The Hamiltonian of such systems consists of discrete spins each one of which can take more than two values or Ising spins with multi-spin interactions. The spectrum of possible cycles in simple examples is examined. The relevance of these results to systems with finite zero-temperature entropy and to the class of undecidable problems is also discussed.