The Garden of Eden Theorem for cellular automata and for symbolic dynamical systems

We survey the most recent and general results on Garden of Eden (briefly GOE) type theorems in the setting of Symbolic Dynamical Systems and Cellular Automata. We present a GOE type theorem for (one-dimensional) irreducible sub-shifts of finite type and show that a generalization to sofic shifts does not hold in general. We also present a detailed and self-contained proof of a GOE type theorem of Gromov for maps of bounded propagation, between strongly-irreducible stable spaces of finite type over amenable graphs, admitting a dense pseudogroup of holonomy maps.