= Parabolic disk for z minus z squared
{title2=$\Delta=\{|z-1/2|<1/2\},\quad |P^n(z)|<1/(n+1)$}
For $P(z)=z-z^2$, the disc $\Delta=\{|z-1/2|<1/2\}$ corresponds under $w=1/z$ to $\Re w>1$. The transformed map is $T(w)=w+1+1/(w-1)$, so $\Re T(w)>\Re w+1$. Thus $P(\Delta)\subseteq\Delta$ and $|P^n(z)|<1/(n+1)$ there. The <Fatou component> $U$ containing the disc has $P^n\to0$ locally uniformly by normality and the identity theorem. It is forward invariant. The point $1/4$ in the disc has the entire fiber $\{1/2\}$, so the <Fatou component containing an entire fiber is completely invariant> criterion gives $P^{-1}(U)=U$. Zero itself is a Julia point: a normal subsequential limit near zero would be zero by the disc limit, but every iterate has derivative one at zero, contradicting derivative convergence.
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