Since multiplier l = f '(z_{*}) =
2z_{*} a fixed point is attracting if
l < 1, z_{*} < 1/2 i.e. z_{*} lies inside the u = 1/2 exp(if) circle. It follows from (*) that c = z_{*}  z_{*}^{2} and corresponding c lies inside the cardioid c = u  u^{2} = 1/2 exp(if)  1/4 exp(2if), Re(c) = [cos(f)  cos(2f) / 2] / 2, Im(c) = [sin(f)  sin(2f) / 2] / 2. For all c inside the main cardioid, the critical z_{o} = 0 orbit is attracted to fixed point z_{*} . 
We get one more usefull "square" parametrization if we use
c = 1/4  a^{2}. As a^{2} = (a)^{2}
the M is symmetrical with respect to a = 0. After substitution of
c = 1/4  a^{2} into (*) we get:
z_{*} = 1/2 + a. z_{*} is attracting if 1/2 + a < 1/2, i.e. a lies inside one of the circles e^{if} /2 + 1/2 . 
Point on the main cardioid corresponding to an internal angle
f = 2p m/n lies at
c_{f} = e^{if}/2  e^{2if}/4 . In "square" parametrization

The second factor has two roots
z_{3,4} = 1/2 + (3/4  c)^{1/2}. These two roots form period2 orbit. Since z_{3} z_{4} = c + 1 the multiplier of the orbit is l = f '(z_{3}) f '(z_{4}) = 4z_{3} z_{4} = 4(c + 1). Therefore the orbit is attracting while c + 1 < 1/4 or c lies within the [exp(if) / 4  1] circle. This is exactly equation of the biggest 1/2 bulb to the left of the main cardioid. I.e. the main cardioid and the 1/2 bulb are connected and touch each other in one point z = 3/4. 
You see the points z_{14} positions
for c = 0.71+0.1i (inside the main cardioid).
Since z_{3} + z_{4} = 1/2 the roots are
symmetrical with respect to the point z = 1/2.
We will watch fixed points and periodic orbits movement in more detail on
the next pages.
Repeller z_{2} lies in Julia set. Is it "very often" the extreme right point for connected Js ("very often" because it is not true e.g. for "cauliflower"). 