Hello MathOverflow, my first question,
I apologize for the LaTeX, it works in preview but not in Safari once posted.
There are many methods out there that generate a list of unique unit fractions that sum to some rational number. One of the simplest is called the "Splitting Algorithm" which uses the identity $\frac{1}{a}=\frac{1}{a+1}+\frac{1}{a(a+1)}$. Any fraction $\frac{a}{b}$ can be represented as $\sum^a \frac{1}{b}$. The method looks at all the fractions that are duplicates, keeps one of them, and applies the identity again.
For example
$\frac{2}{3}=\frac{1}{3}+\frac{1}{3}=\frac{1}{3}+\frac{1}{4}+\frac{1}{12}$
Each of the denominators can be looked at as a composition of functions $f(a)=a+1$ and $g(a)=a(a+1)$. The method works in spite of $f,g,f\circ g,g\circ f,\cdots$ are never equal. fractions like $\frac{5}{2}$. When the method is applied to fractions like this duplicates will appear although it has been proved they can be removed by subsequent applications of the method so that no remaining compositions are equal.
The Question
Can you think of any other areas where compositions of functions are used like this? I know that on its own that is too general; used in the list of sums, products, useful identities, number theory? The best case scenario would be that there are other problems like this and they can be searched for under a common name.
Edit: method produces unique unit fractions; clarified 'equal', when I wrote that I was thinking of cases like $\frac{b-1}{b}$ that may not lead to duplicates
Best Answer
Computing a continued fraction representation for a real number x can be seen as a repeated application of two functions. Starting with some real number x in [0,1[, apply 1/x, then x+1 the correct amount of time to come back in [0,1[, then 1/x again and so on.
The theory of fuchsian groups makes use of these codings to connect geometric properties of hyperbolic spaces to arithmetic properties of real numbers. Continued fractions representation for example is related to the geodesics on the modular surface $SL_2(R)/SL_2(Z)$.