On the Surjunctivity and the Garden of Eden Theorem for Non-Uniform Cellular Automata

Non-uniform cellular automata (NUCA) are an extension of cellular automata with multiple local rules in different cells. We show that if the distribution of local rules is uniformly recurrent, or recurrent in the one-dimensional case, the Garden of Eden theorem holds. We also show that for any one-dimensional non-recurrent distribution, there is a substitution of local rules that defines a NUCA which does not satisfy the Garden of Eden theorem. Finally, we show that a one-dimensional rule distribution asymptotic to recurrent distribution defines a surjunctive NUCA.