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A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit. (English) Zbl 1315.82005
Summary: We provide an example of a discrete-time Markov process on the three-dimensional infinite integer lattice with \(\mathbb{Z}_q\)-invariant Bernoulli-increments which has as local state space the cyclic group \(\mathbb{Z}_q\). 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.

82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
68Q80 Cellular automata (computational aspects)
Full Text: DOI arXiv
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