 It has been called "one of the fundamental results in symbolic dynamics".The theorem states that a function from a shift space to itself represents the transition function of a one-dimensional cellular automaton if and only if it is continuous (with respect to the Cantor topology) and equivariant (with respect to the shift map).More generally, it asserts that the morphisms between any two shift spaces (i.e., continuous mappings that commute with the shift) are exactly those mappings which can be defined uniformly by a local rule.