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A non-ergodic probabilistic cellular automaton with a unique invariant measure

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  • Chassaing, Philippe
  • Mairesse, Jean

Abstract

We exhibit a Probabilistic Cellular Automaton (PCA) on { 0 , 1 } Z with a neighborhood of size 2 which is non-ergodic although it has a unique invariant measure. This answers by the negative an old open question on whether uniqueness of the invariant measure implies ergodicity for a PCA.

Suggested Citation

  • Chassaing, Philippe & Mairesse, Jean, 2011. "A non-ergodic probabilistic cellular automaton with a unique invariant measure," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2474-2487, November.
  • Handle: RePEc:eee:spapps:v:121:y:2011:i:11:p:2474-2487
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    Cited by:

    1. Casse, Jérôme & Marckert, Jean-François, 2015. "Markovianity of the invariant distribution of probabilistic cellular automata on the line," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3458-3483.
    2. Jahnel, Benedikt & Külske, Christof, 2015. "A class of non-ergodic probabilistic cellular automata with unique invariant measure and quasi-periodic orbit," Stochastic Processes and their Applications, Elsevier, vol. 125(6), pages 2427-2450.

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