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 - u2 = 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 zo = 0 orbit is attracted to fixed point z* .
We get one more usefull "square" parametrization if we use
c = 1/4 - a2. As a2 = (-a)2
the M is symmetrical with respect to a = 0. After substitution of
c = 1/4 - a2 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
eif /2 +- 1/2 .
Point on the main cardioid corresponding to an internal angle
f = 2p m/n lies at
cf = eif/2 - e2if/4 .
In "square" parametrization
The second factor has two roots
z3,4 = -1/2 +- (-3/4 - c)1/2.
These two roots form period-2 orbit. Since z3 z4 = c + 1 the multiplier of the orbit is
l = f '(z3) f '(z4) = 4z3 z4 = 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 z1-4 positions
for c = -0.71+0.1i (inside the main cardioid).
Since z3 + z4 = -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 z2 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").