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(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!

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up vote 1 down vote accepted

Try to plot $f^n(x)$ for a few values of $n$.

We have the following for the first couple:

enter image description here for $n=1$ enter image description here for $n=2$ enter image description here for $n=3$

And superimposed we have the following:

enter image description here

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:

enter image description here

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

enter image description here

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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Is there a systematic way of doing this other than calculating the iterates and graphing them, and looking for more than 2 points where fn(x)=x? –  tamefoxes Jan 31 '13 at 4:11
    
if $T(x) = \mu \min(x,1-x)$, then look at values of $\frac{\mu r}{\mu^n+1}$ for orbits of period $n$ and integers $r$. There isn't a general solution, I don't think, but there are some things you can say. –  Sam DeHority Jan 31 '13 at 4:47
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