A garden of Eden theorem for linear subshifts (Q2884075)

The Moore-Myhill `garden of Eden' theorem is extended to the setting of linear cellular automata over linear shifts of finite type. For an amenable group \(G\) and any field \(K\) it is shown that if \(X\subset (K^d)^G\) for some \(d\geq1\) is a strongly irreducible linear shift of finite type then a linear cellular automaton \(X\to X\) is surjective if and only if it is pre-injective. Some related examples and results are also given, and it is shown that if \(G\) is countable and \(X\subset(K^d)^G\) is a strongly irreducible linear subshift, then any injective linear cellular automaton \(X\to X\) is surjective.