A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science Part VII: ISLES of Eden

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 30 and 110 . 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,