What Are Functional Powers and How Can We Explore f^n(x)?

[willsc8forwin]

I was reading on wikipedia about "functional powers", but I can't seem to find anything on it outside of this one section. I was wondering if there's any way to show anything for f^n(x). This is more of a general plea for more information on the topic than a specific question.

Oh and here's the wikipdia page I mentioned.http://en.wikipedia.org/wiki/Function_composition#Functional_powers"

 

[micromass]

I was reading on wikipedia about "functional powers", but I can't seem to find anything on it outside of this one section. I was wondering if there's any way to show anything for f^n(x). This is more of a general plea for more information on the topic than a specific question.

Oh and here's the wikipdia page I mentioned.http://en.wikipedia.org/wiki/Function_composition#Functional_powers"

Functional powers are used for example in Banach fixed point theorem http://en.wikipedia.org/wiki/Banach_fixed_point_theorem where one take a function f and investigates the limit of

$$x,f(x),f(f(x)),f(f(f(x))),...$$

Also check the theory of Markov chains, dynamics systems and fractals (i.e. iterated function systems). This used functional powers.

 

[Tac-Tics]

Fixpoints are also very relevant in type theory and functional programming. The fixpoint function fix f = f(f(f(...))) allows for general recursion (which is useful for programmers), but opens the possibility of nonterminating programs (infinite loops) which correspond under the Curry-Howard isomorphism to proofs of false propositions.