I'm currently going through Alan Beardon's book "Iteration of Rational Functions" and I'm a little stuck on his explanation of Corollary 9.5.3. which states that "If every finite critical points of a polynomial is preperiodic, then the Fatou set is connected and simply connected".
I understand that if the Fatou set is not connected, then there must exist a periodic Fatou component that contains a finite critical point and isn't $F_\infty$. However apparently, the assumption that all critical points are preperiodic gives us a contradiction here.. but I'm not sure how.
If anyone could break down his argument, it would be greatly appreciated.
