Chapter 6: Starting from Randomness

Section 7: The Notion of Attractors

Nested structure of attractors

Associating with each sequence of length n (and k possible colors for each element) a number Sum[a[i] k-i, {i, n}], the set of sequences that occur in the limit n -> ∞ forms a Cantor set. For k = 3, the set of sequences where the second color never occurs corresponds to the standard middle-thirds Cantor set. In general, whenever the possible sequences correspond to paths through a finite network, it follows that the Cantor set obtained has a nested structure. Indeed, constructing the Cantor set in levels by considering progressively longer sequences is effectively equivalent to following successive steps in a substitution system of the kind discussed on page 83. (To see the equivalence first set up s kinds of elements in the substitution system corresponding to the s nodes in the network.) Note that if the possible sequences cannot be described by a network, then the Cantor set obtained will inevitably not have a strictly nested form.

From Stephen Wolfram: A New Kind of Science [citation]