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

Benedikt Jahnel; Christof Kuelske

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arXiv · 1404.3314

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

Benedikt Jahnel·Christof Kuelske

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Abstract

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.

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Benedikt Jahnel, Christof Kuelske. 2014-09-20. A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit. https://arxiv.org/abs/1404.3314

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