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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
up vote 4 down vote accepted

This is usually called Fatou's theorem, I think. Good references for it are

  • Milnor, Dynamics in one complex variable, 3rd ed. Theorem 8.6
  • Carleson and Gamelin, Complex Dynamics, Theorem III.2.2

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.

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So it applies to any "rational map" - i.e., any function which is the ratio of two complex-valued polynomials? – MathematicalOrchid May 7 '12 at 18:20
That's right. There are some other situations where an analogous theorem applies. For instance, if you define everything correctly it holds for nonivertible holomorphic maps on higher dimensional projective spaces as well, but this is harder. Interestingly enough, the theorem fails if you work over nonarchimedean fields. – froggie May 7 '12 at 18:34
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

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