A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit

We provide an example of a discrete-time Markov process on the three-dimensional infinite integer lattice with Z_q-invariant Bernoulli-increments which has as local state space the cyclic group Z_q. We show that the system has a unique invariant measure, but remarkably possesses an invariant set of measures on which the dynamics is conjugate to an irrational rotation on the continuous sphere S^1. The update mechanism we construct is exponentially well localized on the lattice.