Post Critical Set

Post Critical Set (of order T) Button

The "Post Critical Set of order T" button will enable the user to generate a set labeled "Post Critical set of order T" from the selected INPUT FUNCTIONS. Here, the Post Critical set is defined to be the set of all (finite) critical values of any map which can be made from composing any of the selected INPUT FUNCTIONS any number of times (thus it is the set of all critical values of all maps in the semigroup generated by the selected INPUT FUNCTIONS). Since the number of such compositions is clearly infinite (as it must contain all infinitely many iterates of any one of the selected INPUT FUNCTIONS) we can only ask the computer to approximate this set. And so the "Post Critical set of order T" will be the critical values of all maps which can be written as a composition of no more than T selected maps, allowing for duplicates. In other words, this is the set of critical values of the maps whose "word length" (defined by the number of, not necessarily distinct, INPUT FUNCTIONS composed to make up the map) is no more than T. Thus, if the user sets T=1, then the "Post Critical set of order T=1" will simply be the set of critical values of the selected INPUT FUNCTIONS.

The computer algorithm for computing the "Post Critical set of order T" is as follows:

Call the selected INPUT FUNCTIONS by the names G₁,...,G_k.
The program will then compute sets V₁,...,V_k as follows:

First, we will call the corresponding sub-generator functions f₁,...,f_k. So, e.g., G₁ is the function f₁ iterated m₁-times. See here for an example of a sub-generator and corresponding INPUT FUNCTION (generator function).

If f₁ is a cubic function of the form az³ + b, with a ≠0 (whose sole critical point is at z = 0), then the set V₁ should contain the m₁ points

f₁(0), f₁( f₁(0)), ..., f₁^m1(0).

If f₁ is a quadratic function of the form az² +bz + c, with a ≠0, (whose sole critical point is at z = -b/(2a)), then the set V₁ should contain the m₁ points

f₁(-b/(2a)), f₁(f₁(-b/(2a))), ..., f₁^m1(-b/(2a)).

If f₁ is any other kind of function, then the set V₁ should be empty since the map G₁has no critical points.

The set V₂ should be determined the exact same way using the function G₂. Similarly, for the other sets up to V_k.

Then the set W₁ = V₁∪...∪V_k will be created. If W₁ is empty, then the user will be notified that "The selected functions have no critical points."

If W₁ is not empty, then a set W₂ = G₁(W₁)∪ G₂(W₁)∪...∪G_k(W₁) will be created.

Then a set W₃ = G₁(W₂)∪ G₂(W₂)∪...∪G_k(W₂) will be created.

Continue in this fashion to create sets W₁ , W₂ , ..., W_T.

Then the Output Set created and labeled "Post Critical set of order T" is the set P = W₁ ∪...∪W_T.

Note that this process does not use any information about Points to Plot, Skips, Seed Value, Set Type or Process Type.