6

I want to prove that the immediate basin of attraction of a finite attracting fixed or periodic point is simply connected. We are talking about complex numbers ! According to Remark 2 p. 281 and Exercise 4.2 p. 283 of the text of Devaney [1],

If $z_0$ is a finite attracting orbit (i.e., $z_0 \neq +\infty$), then any component of its basin of attraction is simply connected. This fact is an easy consequence of the Maximum Principle (see Exercise 4.2).

Exercise 4.2. Prove that the immediate attracting basin of a (finite) attracting periodic point is simply connected.

Apparently easy so I must be overlooking something. Who can give me an accurate proof ?

[1] Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed., Westview Press, 2003.

Rodrigo de Azevedo
  • 23,223
Cor Gaajetaan
  • 61
  • By definition, the immediate basin of attraction is the connected component of the basin of attraction containing $P$ and so is connected. Your question is about it being simply connected. You may want to improve the title to reflect this. – lhf Nov 30 '16 at 11:29

1 Answers1

3

If you want to be rigorous then I don't think it is that easy (but I am not a real expert in the domain). One possible approach is to use two topological results:

Lemma 1: If $A\subset \widehat{\Bbb C}$ (the Riemann sphere) is connected then each connected component of $ \widehat{\Bbb C}\setminus A$ is simply connected. (more or less obvious)

You may use this in the following way: Let $\gamma: [0,1]\rightarrow {\Bbb C}$, $\gamma(0)=\gamma(1)$ be a continuous map (a loop). Let $\Omega\subset \widehat{\Bbb C}$ be the connected component of $\widehat {\Bbb C}\setminus \gamma$ containing $\infty$. We define $\Phi(\gamma) = \widehat{\Bbb C}\setminus \Omega$ to be its compliment. By the above lemma, both $\Omega$ and (more interestingly here) $\Phi(\gamma)\subset {\Bbb C}$ are simply connected. Furthermore, $\partial \Phi(\gamma) \subset \gamma$. The set $\Phi(\gamma)$ contains intuitively the points "encircled by $\gamma$".

The second topological result comes from the maximum principle.

Lemma 2: If $D\subset {\Bbb C}$ is a bounded domain and $f:{\Bbb C}\rightarrow {\Bbb C}$ is analytic then $\partial f (D) \subset f (\partial D)$, i.e. the image of the boundary contains the boundary of the image.

To see this, note that if $w_0=f(z_0)\in \partial f(D)$, $z_0\in D$ but $w_0$ is not in $f(\partial D)$ then you get a contradiction with the maximum principle for the function $z\in D \mapsto 1/(f(z)-w)$ by choosing $w$ close enough to $w_0$ but in the complement of $f(D)$.

Now, let $p$ be an attractive fixed point of a periodic point of an entire map $f:{\Bbb C}\rightarrow {\Bbb C}$ (or some iterate of it in the case of a periodic point). Let $\gamma$ be a loop consisting of points in a basin of attraction of $p$ (possibly the immediate basin but it need not be) and define as above $D=\Phi(\gamma)$.

By the two Lemmas $$ \partial f^n (D) \subset f^n(\partial D) \subset f^n(\gamma)$$ which implies that $f^n(D)$ converges to $p$ since $f^n (\gamma)$ does. Thus, $D$ belongs to the same basin of attraction as $\gamma$ and since $D$ is simply connected so is the basin.

H. H. Rugh
  • 35,992
  • Your explanation of lemma 2 (to see this....) is to compact for me. I tried to write our a complete proof, but I don't see the crucial contradiction. I can follow the overall idea of your complete answer but I really wonder is this is what Devaney meant by "EASY consequence" in his text. – Cor Gaajetaan Sep 18 '17 at 08:28
  • Choose $w$ in the complement of $f(D)$ so that $0<|w_0-w|<r < R < |f(z)-w|$ for all $z\in \partial D$. Then $g(z)=1/(f(z)-w)$ is analytic on $\bar{D}$ but $|g(z_0)|> 1/r > 1/R > \max{ |g(z)|: z\in \partial D}$ contradicting the max principle. There are btw other easy ways to see this: A non-constant analytic map is "open" so is $B(z,r)\subset D$ then $f(B(z,r))$ is an open neighborhood of $f(z)$ which therefore can not be a boundary point. Devaney's use of 'easy' is perhaps not so appropriate in this context (unless he has proven something similar before in the book?) – H. H. Rugh Sep 18 '17 at 11:12
  • In my view Devaney did'n't proof anything similar, so your proof is the definite answer to my question ! – Cor Gaajetaan Sep 22 '17 at 07:46
  • I also had a look in Carleson-Gamelin "Complex dynamics". They also just state this result somewhere as a consequence of the max principle without further details. A bit fast, in my opinion. – H. H. Rugh Sep 22 '17 at 08:10
  • I just bought that book ! Can you give me the pages where you looked ? Maybe a nice start for me to continue my study of complex dynamics ! – Cor Gaajetaan Sep 23 '17 at 09:52
  • The book is very nice but not that easy. In my 1993 edition, there are special cases mentioned in chap III p 55 3rd example and p 59 Thm 2.2 (this one avoids the max principle through other arguments), and then in chap IV p 70 Thm 1.1 (which b.t.w. mentions the content of my Lemma 1, without proof) and p 71 Thm 1.3 (of Sullivan) where more or less the full statement appears (with few details, as it is just preliminary to proving something else). – H. H. Rugh Sep 23 '17 at 11:59