Weak Gardens of Eden for 1-dimensional tessellation automata (Q1068546)

If T is the parallel map associated with a 1-dimensional tessellation automaton, then we say a configuration f is a weak Garden of Eden for T if f has no pre-image under T other than a shift of itself. Let WG(T) \(= the\) set of weak Gardens of Eden for T and G(T) \(= the\) set of Gardens of Eden (i.e., the set of configurations not in the range of T). Typically members of WG(T)-G(T) satisfy an equation of the form \(Tf=S^ mf\) where \(S^ m\) is the shift defined by \((S^ mf)(j)=f(j+m)\). Subject to a mild restriction on m, the equation \(Tf=S^ mf\) always has a solution f, and all such solutions are periodic. We present a few other properties of weak Gardens of Eden and a characterization of WG(T) for a class of parallel maps we call (0,1)-characteristic transformations in the case where there are at least three cell states.