Abstract: This paper continues our quest to develop a rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulli στ-shift rules and eight hyper Bernoulli στ-shift rules, the latter including such famous rules 
and 
. All exhibit a bizarre composite wave dynamics with arbitrarily large Bernoulli velocity σ and Bernoulli return time τ as the length L → ∞.
Basin tree diagrams of all ten complex Bernoulli στ-shift rules are exhibited for lengths L = 3, 4, …, 8. Superficial as it may seem, these basin tree diagrams suggest general qualitative properties which have since been proved to be true in general. Two such properties form the main results of this paper; namely,
Explicit global state transition formulas are given for local rules
,
and
. Such formulas led to the rigorous proof of several surprising
periodicity constraints for rule
, and to the discovery of a new global,
quasi-equivalence class, defined via an
alternating transformation. In particular, local rules
and
are globally quasi-equivalent where corresponding space-time patterns can be derived from each other by simply complementing every other row.
Another important result of this paper is the discovery of a scale-free phenomenon exhibited by the local rules 
, 
and 
. In particular, the period "T" of all attractors of rules 
, 
and 
, as well as of all isles of Eden of rules 
and 
, increases linearly with unit slope, in logarithmic scale, with the length L.
