A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART VIII: MORE ISLES OF EDEN

This paper presents the basin tree diagrams of all hyper Bernoulli σ_τ-shift rules for string lengths L = 3, 4, …, 8. These diagrams have revealed many global and time-asymptotic properties that we have subsequently proved to be true for all L < ∞. In particular, we have proved that local rule has no Isles of Eden for all L, and that local rules and are inhabited by a dense set (continuum) of Isles of Eden if, and only if, L is an odd integer. A novel and powerful graph-theoretic tool, called Isles-of-Eden digraph, has been developed and can be used to test the existence of dense Isles of Eden of any local rule which satisfies certain constraints, such as rules , , , , as well as all invariant local rules, such as rules , , and , subject to no constraints.