# Every basin of attraction contains a critical point?

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.)

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Basin of attraction of what? –  Mariano Suárez-Alvarez May 7 '12 at 17:44
I really liked the book Chaos: An Introduction to Dynamical Systems by Yorke, Alligood and Sauer. It think it will answer this question and many others that you might have about similar matters. –  MJD May 7 '12 at 17:49
N.B. "Gaston" is the given name. "Julia" is the surname. :) –  Ｊ. Ｍ. May 7 '12 at 18:14
@J.M. I guess I'm feeling retarded today. Obviously I meant Julia and/or Fatou. >_< –  MathematicalOrchid May 7 '12 at 18:17
PS. Is "dynamical system" the correct term for this sort of thing? (I.e., where we compute a series of values by repeatedly applying a deterministic function.) I'm never sure exactly what title this comes under... –  MathematicalOrchid May 7 '12 at 18:22

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.
OK. So it wouldn't work for, say, some trigonometric function like $\sin z$, but it would work for something like Newton's iteration on solving a polynomial. Does any similar result hold for dynamical systems not involving complex numbers? –  MathematicalOrchid May 7 '12 at 18:38
I don't know much about iteration of transcendental functions like $\sin z$ so I couldn't really say. But it definitely works for rational maps over the complex numbers. I don't know of any general theorems similar to this over not the complex numbers, but maybe given a specific kind of dynamical system you can say something? Maybe other people will have some input.... –  froggie May 7 '12 at 19:27