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 ZqZq-invariant Bernoulli-increments which has as local state space the cyclic group ZqZq. 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 S1S1. The update mechanism we construct is exponentially well localized on the lattice.