Parabolic basin

A parabolic basin consists of points that eventually enter an attracting direction of a parabolic cycle. For a suitable return iterate, an attracting petal is a forward-invariant open region with the periodic point on its boundary, and the return iterates tend to that point there. The union of its inverse images is an open parabolic basin contained in the Fatou set; the Fatou components meeting the attracting petals form the immediate basin. Its boundary lies in the Julia set.

The parabolic cycle itself and points that land exactly on it belong to the Julia set, so mere pointwise convergence to that cycle is not the definition of the open parabolic basin. For $f(z)=z+z^2$, zero is a fixed point with periodic-orbit multiplier one. Small negative real starting values approach zero without reaching it, whereas zero itself is a Julia set point. The reflected example parabolic disk for z minus z squared gives an explicit forward-invariant region and a quantitative convergence bound.^[ref]

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 $P^n\to 0$ 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.