Simple connectedness of basin of attraction 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.
 A: 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.
