Iteration of maps by an automaton

Abstract Let x =( x 1 , x 2 ,…) with the x i elements of afinite set A be an automatic sequence, i.e., x is generated by a tag machine with deletion number one. Let F be a finite set and suppose maps ϕ a : F → F are given for all aϵA . It is shown constructively that the sequence of iterates (ϕ x1 , ϕ x2 o ϕ x1 , ϕ x3 o ϕ x2 o ϕ x1 ,…) is again an automatic sequence.