05/02/2009, 02:23 AM
(This post was last modified: 05/12/2009, 02:12 AM by Base-Acid Tetration.)

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:

(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: