Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions

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For a general attractive Probabilistic Cellular Automata on S Z d, we prove that the (time-) convergence towards equilibrium of this Markovian parallel dynamics, exponentially fast in the uniform norm, is equivalent to a condition (A). This condition means the exponential decay of the inuence from the boundary for the invariant measures of the system restricted to nite boxes. For a class of reversible PCA dynamics on --1, +1 Z d, with a naturally associated Gibbsian potential φ, we prove that a (spatial-) weak mixing condition (WM) for φ implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to φ holds. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition.