You see these small circles around the attracting fixed points below (here |λ| = 0.832). Inverse function f -1(z) maps the disks vice versa. One can extend the map analitically , while f -1(z) is a smooth nonsingular function with finite derivative.
We can continue fc-1 up to the outside border of the yellow region. The border contains the point z = c and is mapped to the figure eight curve with the critical point zc in the center. Therefore iterations fcon(zc ) converge to z∗ for large n (the orbit is called the critical orbit). This is the subject of the Fatou theorem.
Fatou theorem: every attracting cycle for a polynomial or rational function attracts at least one critical point.
As since quadratic maps have the only critical point zc = 0 then quadratic J may have the only finite attractive cycle! (There is one more critical point at infinity which attracts diverging orbits.) Thus, testing the critical point shows if there is any finite attractive cycle.
 John W. Milnor "Dynamics in One Complex Variable" § 8.5