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

We exhibit a Probabilistic Cellular Automaton (PCA) on {0,1}Z{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.

► Study of the behavior of probabilistic cellular automata (PCA). ► First example of a PCA with a unique but non-attractive invariant measure. ► Exact computation of the density evolution for annihilating particles PCA. ► Exact computation of the density evolution for coalescing particles PCA.