TITLE: Finding an example of a periodic point with prime period strictly greater than 2.
QUESTION [1 upvotes]: (iv) Give an example of a periodic orbit with prime period strictly greater than 2.
The function that I am given is: $T(x) = \begin{cases}\frac{3}{2}x &x \leq .5\\
                                        \frac{3}{2}(1-x) & x \ge .5\end{cases}$
I already found the second iterate of $T(x)$ but I am having a hard time trying to construct the third iterate of $T(x)$ assuming that it is what it is required to find periodic points with prime period greater than 2. 
Any help is appreciated!

REPLY [2 votes]: Try to plot $f^n(x)$ for a few values of $n$. 
We have the following for the first couple:
 for $n=1$
 for $n=2$
 for $n=3$
And superimposed we have the following:

Notice that each peak basically "splits" into two peaks. This should be intuitive from the function definition, and give you some idea for what $f^n$ is like for an arbitrary $n$. 
But if we are trying to find a periodic orbit of period 3, i.e. a fixed point of $f^3(x)$, graph $f^3(x)$ superimposed with $f(x) = x$ to give:

This shows that there is exactly one non-fixed period 3 orbit of $f$, because there is one point where $f^3(x) = x$. 
This value happens to be $x = \frac{3}{5}$, but you'll notice that
$$f( \frac{3}{5}) = \frac{3}{5}$$
So this isn't an orbit of order strictly greater than 2. The same thing happens for $n=5$. For $n=7$ however we get the far more complcated 

Which has one solution of $x = \frac{894}{2315}$, giving the desired orbit. 
$$\frac{894}{2315},\frac{1461}{2315},\frac{1281}{2315},\frac{1551}{2315},\frac{1146}{2315},\frac{1719}{2315},\frac{894}{2315}$$
This map belongs to the more general class of functions called tent maps, and the wikipedia page gives a number of interesting behaviors, notable that they exhibit a bifurcation structure similar to the map $x^2 +c$, and the logistic map, because the logistic map and the tent map are  topologically conjugate.
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