Prevalence and structure of adding machines for cellular automata

Consider the collection of left permutive cellular automata Φ with no memory, defined on the space S of all doubly infinite sequences from a finite alphabet. There exists Z ˜ , a dense subset of S , such that Φ : cl { Φ n ( x ) : n ⩾ 0 } → cl { Φ n ( x ) : n ⩾ 0 } is topologically conjugate to an odometer for all x ∈ Z ˜ so long as Φ m is not the identity map for any m. Moreover, Φ generates the same odometer for all x ∈ Z ˜ . The set Z ˜ is a dense G δ subset with full measure of a particular subspace of S .