How many functions satisfy $f_7(x)=x$?

Question

Define a function $f(x):=4x(1-x)$. How many $x$ in $[0,1]$ satisfy $$f(f(f(f(f(f(f(x)))))))=x$$

I tried to do it by plugging in $x=\frac{1}{y}$ considering $x\in (0,1]$ and then proceed by considering the $x=0$ case separately. But I got to nowhere.

Answer 1

If $x = (1-\cos(t))/2$, $f(x) = (1 - \cos(2t))/2$. Thus we need $\cos(t) = \cos(128 t)$, which says $128 t = \pm t + 2 n \pi$ for some integer $n$, i.e. $t = \frac{2n\pi}{127}$ or $t = \frac{2n\pi}{129}$. All values of $x$ are obtained with $t$ in $[0,\pi]$, and thus $t = \frac{2n\pi}{127}$ for $0 \le n \le 63$ or $t = \frac{2n \pi}{129}$ for $0 \le n \le 64$. $t=0$ is common to both, so there are $128$ solutions.