Ergodicity of PCA: Equivalence between Spatial and Temporal Mixing Conditions

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 influence from the boundary for the invariant measures of the system restricted to finite boxes. For a class of reversible PCA dynamics on $\{-1, +1\}^{Z^d}$ with a naturally associated Gibbsian potential $\varphi$, we prove that a (spatial-) weak mixing condition (WM) for $\varphi$ implies the validity of the assumption (A); thus exponential (time-) ergodicity of these dynamics towards the unique Gibbs measure associated to $\varphi$ holds. On some particular examples we state that exponential ergodicity holds as soon as there is no phase transition