Here is a question I am not able to answer.
Find the limit points of the iterative sequence given by $x_0=2$ and $x_{n+1}=2\cos(x_n)$
If one relaces $2\cos$ by $\cos$ then it is a classical exercise and the sequence converges. Here $x_n$ s bounded so has limit values which form a compact set $A \subset [-2,2]$. Numerical simulations suggest that there exists $a,b,c>0$ such that $A=[-a,b] \cup [c,2]$. But I am not able to prove it.
Well the only thing I can tell is that the $y$ such that $2\cos (y)=y$ is such that $2\sin (y) >1$ and thus there is a region around $y$ which can not contain any limit value.