Weak gardens of Eden for 1-dimensional tessellation automata

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 W G ( 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 W G ( T ) − G ( T ) satisfy an equation of the form T f = S m f where S m is the shift defined by ( S m f ) ( j ) = f ( j + m ) . Subject to a mild restriction on m , the equation T f = S m f 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 W G ( T ) for a class of parallel maps we call ( 0 , 1 ) -characteristic transformations in the case where there are at least three cell states.