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Functional super-iteration and hierarchy of functional hyper-iterations
We all know the functional "exponentiation" or "power" for functional iteration,
(I use F^circle n, f^n (x), or f^circle n (x) if clarity is needed, for iteration)
That means we can do this for some function:
where the function f is iterated g(x) times at x.

We could investigate the properties and results of functional "tetration", or super-iteration. The "upper-left exponent" notation can be abuse, or use the circle notation like

Initial Definition: (might have gotten my variables wrong)
Given a and a
a functional super-iteration is defined by the recurrence relation:

Already a hierarchy of operations on functions that is much like the arithmetic hyper-operations is evident:

Function composition:
Function iteration: A function can be iterated a constant times, or by g(x) times.
Functional superiteration:

Messages In This Thread
Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/02/2009, 02:23 AM

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