  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.
