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

2010/09/01 by Philippe Chassaing, Jean Mairesse, Chassaing, Philippe +1 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Cellular Automata and Applications #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.1009.0143

openalex publication_date 2010/09/01 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

We exhibit a Probabilistic Cellular Automaton (PCA) on the integers with an\nalphabet and a neighborhood of size 2 which is non-ergodic although it has a\nunique invariant measure. This answers by the negative an old open question on\nwhether uniqueness of the invariant measure implies ergodicity for a PCA.\n

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