Note (c) for Iterated Maps and the Chaos Phenomenon…A particularly well-studied example (see page 918 ) is the so-called logistic map x  a x (1 - x) . The base 2 digit sequences obtained with this map starting from x = 1/8 are shown below for various values of a . The quadratic nature of the map typically causes the total number of digits to double at each step.

Note (a) for Iterated Maps and the Chaos Phenomenon…Mathematical perspectives [on iterated map complexity] Mathematicians may be confused by my discussion of complexity in iterated maps.

Note (b) for Iterated Maps and the Chaos Phenomenon…Exact iterates [in iterated maps] For any integer a the n th iterate of x  FractionalPart[a x] can be written as FractionalPart[a n x] , or equivalently 1/2 - ArcTan[Cot[a n π x]]/ π .

Note (c) for Iterated Maps and the Chaos Phenomenon…(An example is NestList[Mod[2 #, 1]&, N[ π /4, 40], 200] ; Map[Precision, list] gives the number of significant digits of each element in the list.) … As an example, consider the iterated map x  Mod[2x, 1] discussed in the main text. At each step, this map shifts all the base 2 digits in x one position to the left.

[No text on this page] Maps of where in the space of parameters for the substitution systems on the facing page the patterns obtained overlap the region indicated in the icon at the top left of each picture. … The maps shown can be thought of as being made by taking an infinitely dense limit of the array of pictures on the facing page , but keeping only what one sees in each picture by looking through a peephole at a particular position relative to the original stem.

And indeed the pictures below illustrate that even in such cases changes in digit sequences are progressively amplified—just like in the shift map case (d). … And indeed, what looking at the shift map in terms of digit sequences shows us is that this phenomenon on its own can make no contribution at all to what we can reasonably consider the ultimate production of randomness. … And in this chapter , we have introduced a similar kind of discreteness into our study of systems based on numbers Differences in digit sequences produced by a small change in initial conditions for the four iterated maps discussed in this section .

Simulating mobile automata Given a mobile automaton like the one from page 73 with rules in the form used on page 887 —and behavior of any complexity—the following will yield a causal-invariant substitution system that emulates it: Map[StringJoin, Map[{"AAABB", "ABABB", "ABAABB"} 〚 # + 1 〛 &, Map[Insert[# 〚 1 〛 , 2, 2]  Insert[# 〚 2, 1 〛 , 2, 2 + # 〚 2, 2 〛 ] &, rule], {2}], {2}]

Generating causal networks If every element generated in the evolution of a generalized substitution system is assigned a unique number, then events can be represented for example by {4, 5}  {11, 12, 13} —and from a list of such events a causal network can be built up using With[{u = Map[First, list]}, MapIndexed[Function[ {e, i}, First[i]  Map[(If[# === {}, ∞ , # 〚 1, 1 〛 ] &)[ Position[u, #]]) &, Last[e]]], list]]

= {} Given a set of sequences of values represented by a particular network, the set obtained after one step of cellular automaton evolution is given by NetCAStep[{k_, r_, rtab_}, net_] := Flatten[ Map[Table[# /. … = {}, AllNet[k], q = ISets[b = Map[Table[ Position[d, NetStep[net, #, a]] 〚 1, 1 〛 , {a, 0, k - 1}]&, d]]; DeleteCases[MapIndexed[#2 〚 2 〛 - 1  #1 &, Rest[ Map[Position[q, #] 〚 1, 1 〛 &, Transpose[Map[Part[#, Map[ First, q]]&, Transpose[b]]], {2}]] - 1, {2}], _  0, {2}]]] DSets[net_, k_:2] := FixedPoint[Union[Flatten[Map[Table[NetStep[net, #, a], {a, 0, k - 1}]&, #], 1]]&, {Range[Length[net]]}] ISets[list_] := FixedPoint[Function[g, Flatten[Map[ Map[Last, Split[Sort[Part[Transpose[{Map[Position[g, #] 〚 1, 1 〛 &, list, {2}], Range[Length[list]]}], #]], First[#1]  First[#2]&], {2}]&, g], 1]], {{1}, Range[2, Length[list]]}] If net has q nodes, then in general MinNet[net] can have as many as 2 q -1 nodes. … To obtain such trimmed networks one can apply the function TrimNet[net_] := With[{m = Apply[Intersection, Map[FixedPoint[ Union[#, Flatten[Map[Last, net 〚 # 〛 , {2}]]]&, #]&, Map[List, Range[Length[net]]]]]}, net 〚 m 〛 /.

Cyclic tag systems [emulating tag systems] From a tag system which depends only on its first element, with rules given as in the note below, the following constructs a cyclic tag system emulating it: TS1ToCT[{n_, subs_}] := With[{k = Length[subs]}, Join[Map[v[Last[#], k] &, subs], Table[{}, {k(n - 1)}]]] u[i_, k_] := Table[If[j  i + 1, 1, 0], {j, k}] v[list_, k_] := Flatten[Map[u[#, k] &, list]] The initial condition for the tag system can be converted using v[list, k] . The list representing the complete history of the resulting cyclic tag system can then be interpreted using Map[Map[Position[#, 1] 〚 1, 1 〛 - 1 &, Partition[#, k]] &, Take[history, {1, -1, n k}]] This construction is relevant to the proof of the universality of rule 110 starting on page 678 .