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But inevitably functions like FixedPoint , ReplaceRepeated and FullSimplify can run into undecidability—so that ultimately they have to be limited by constructs such as $IterationLimit and TimeConstraint .

Iterated maps
For maps of the form x a x (1 - x) discussed on page 920 the attractor for small a is a fixed point, then a period 2 limit cycle, then period 4, 8, 16, etc. There is an accumulation of limit cycles at a ≃ 3.569946 where the system has a special nested structure.

limits oneself to constructing systems out of fairly small numbers of components whose behavior and interactions are somehow simple.
… Indeed, the main difference is just that in engineering explicit human effort is expended to find an appropriate form for the system, whereas in natural selection an iterative random search process is used instead. … And as it turns out, much as we saw in Chapter 7 , this same kind of smooth variation is also what tends to make iterative search methods such as natural selection be successful.

To go further one begins by defining an analog to the Ackermann function of page 906 :
[1][n_] = 2n; [s_][n_] := Nest[ [s - 1], 1, n]
[2][n] is then 2 n , [3] is iterated power, and so on. … And in direct analogy to the transfinite numbers discussed above one can then in principle form a hierarchy of functions using operations like
[ ω + s][n_]:=Nest[ [ ω + s - 1], 1, n]
together with diagonalization at limit ordinals.

Experiments in very restricted situations showed correspondence with iterated maps in which the chaos phenomenon is seen. … I suspect that in the limit where viscosity is fully included most details of initial conditions will simply be damped out, as physical intuition suggests.

In the 1980s, particularly following discoveries in iterated maps and quasicrystals, studies of such spectra were made in the context of number theory and dynamical systems theory. … As suggested by the pictures in the main text, spectra such as (b) and (d) in the limit consist purely of discrete Dirac delta function peaks, while spectra such as (a) and (c) also contain essentially continuous parts. … (c) (Cantor set) In the limit, no single peak contains a nonzero fraction of the power spectrum.

The pattern corresponding to each point is the limit of Nest[Flatten[1 + {c #, Conjugate[c] #}]&, {1}, n] when n ∞ . … A simple way to approximate the pictures in the main text would be to generate patterns by iterating the substitution system a fixed number of times.