Nonergodic Multidimensional System of Automata

Identical stochastic automata having a finite number of states are positioned at all points of a $d$-dimensional integer-valued space. At any instant of discrete time each automaton can go to any one of its states with never-vanishing probabilities depending on its own states and those of a finite number of its “neighbors” at the preceding instant. A system of this type is synthesized which is capable of “remembering” its initial state for an infinitely long time when the system commences operation in one of $n$ distinct states of the type "all automata are in state $k$", where $1\leqslant k\leqslant n$.