Iterating a multiple of sine function makes a square wave So, I found something curious playing around with a graphing calculator. Say we start with a function, $f_1(x) = 2\sin(x)$ and we define a constant, $C$,to be the positive fixed point for $f_1(x)$. Then we talk about the recurrence relation: $f_{n}(x) = f_1 \circ f_{n-1}(x)$. It seems to be the case that: 
$$\lim_{n \to \infty} \frac{1}{C}f_n(x) = \begin{cases}
  \phantom{-}1 & \text{if}\ \sin(x)>0 \\
  -1           & \text{if}\ \sin(x)<0
\end{cases} $$
I toyed with using numbers other than $2$ in this nesting procedure and they all either limit towards horrendous and strange shapes or tend towards being identically zero. Does anyone have any idea as to why this composition creates a square wave, and whether or not other "Fourier-Series" results like this can be found this way?  
 A: This has nothing to do with Fourier series. You get a piecewise constant periodic function;  such functions are also popular examples in Fourier series textbooks; and that's all the connection there is.
Generally, the values of the limit $F(x) = \lim_{n\to\infty} f_n(x)$ are fixed points of $f$, wherever the limit exists. Fixed points can be attracting ($|f'|<1$) or repelling ($|f'|>1$), or indifferent ($|f'|=1$, which is an edge case). Only a discrete set of points converge to a repelling fixed point, because to converge to it, one has to jump directly to that point. So, at almost all points where $F$ is defined, it will be equal to an attracting fixed point of $f$. 
The function $f(x)=2\sin x$ has two attracting points, $\pm C$, and repelling fixed point $0$. Since its range is $[-2,2]$, it suffices to consider the behavior on this interval: 

The positive part of the interval stays invariant (as a whole) under this function, and all points there shift toward the positive attracting point. Hence, $F(x)=C$ when $\sin x>0$. Similarly for the negative part. 
As Rahul observed, this is not a special property of $2$; there is a range of factors $k$ in which $k\sin x$ has the same kind of dynamical behaviour. 
In general, orbits don't need to converge to a fixed point at all; a common scenario is getting trapped within a loop between two or more intervals that get mapped to one another. This happens, for example, with the iteration of $f(x)=2\cos x$, which I wrote about elsewhere. 
