Years and years ago, back when I first became interested in fractals [but didn't know much about anything], I vaguely remember coming across an interesting theorem. The gist of it was that "every basin of attraction contains at least one critical point".
Am I remembering this correctly? Does anybody know any details about it? (E.g., under exactly what circumstances does this theorem apply?) Does this theorem have a name? (I vaguely recall it's meant to be due to Gaston and/or Julia, but that might be wrong.)
Best Answer
This is usually called Fatou's theorem, I think. Good references for it are
The basic idea is this. Suppose that $f\colon \mathbb{P}^1(\mathbb{C})\to \mathbb{P}^1(\mathbb{C})$ is a rational map of degree $d\geq 2$ and that $z_0$ is an attracting fixed point, say with $f'(z_0) = \lambda$ with $0<|\lambda|<1$. Assume for contradiction there is no critical point in the immediate basin of attraction of $z_0$. Because $z_0$ is attracting, there is some small ball $U_0$ around it which lies in the basin of attraction of $z_0$, say with $f(U_0)\Subset U_0$. If $U_0$ does not contain a critical value, then there is some inverse branch $f^{-1}\colon U_0\to U_1$, with $U_0\Subset U_1$. Similarly, if $U_1$ does not contain a critical value, there is an inverse branch $f^{-1}\colon U_1\to U_2$, where $U_1\Subset U_2$. Continuing in this fashion, we can construct inverse branches $f^{-n}\colon U_0\to U_n$ with $U_0\Subset U_n$. Moreover, the $U_n$ don't meet the Julia set of $f$ by construction (they are contained in the basin of attraction of $z_0$). Thus by Montel's theorem there must be a subsequence of the $f^{-n}$ converging on $U_0$. This isn't possible, though, since $(f^{-n})'(z_0) = \lambda^{-n}\to\infty$. Contradiction.