Attracting Fixed point and Period 2 orbit

The main cardioid equation

There are always two fixed points   z∗ = f(z∗ )   for a quadratic map
    f(z∗ ) - z∗ = z∗2 + c - z∗ = 0,   (1)
    z1,2 = 1/2 ∓ (1/4 - c)½.      (2)
A fixed point with multiplier λ = f '(z∗ ) = 2z∗ is attracting if
    |λ| < 1,     |z∗| < 1/2,
i.e. z∗ lies inside the u = ½ exp(iφ) circle. The multiplier on the circle is
    λ = exp(iφ).     (3)
It follows from (1) that c = z∗ - z∗2 and corresponding c lies inside the cardioid
    c = u - u2 = ½ exp(iφ) - ¼ exp(2iφ),
    Re(c) = ½ cos(φ) - ¼ cos(2φ),
    Im(c) = ½ sin(φ) - ¼ sin(2φ).

The M-set in the "quadratic" parametrization

We get one more useful "quadratic" parametrization if we use
    c = 1/4 - a2.     (4)
As since a2 = (-a)2 the M is symmetric with respect to a = 0. After substitution of (4) into (2) we get:
    z∗ = 1/2 ± a.
z∗ is attracting if |1/2 ± a| < 1/2, i.e. a lies inside one of the circles
    ½ eiφ ± 1/2 .

So the (4) transformation converts the main cardioid in two circles.

Internal angles theory

From (3) it follows that if a fixed point lies at the u = ½ exp(i2πm/n) value then under iterations its neighbourhood is rotated by the φ = 2πm/n "internal angle". On the main cardioid the corresponding point lies near the m/n bulb at
    cφ = ½ eiφ - ¼ e2iφ .
In the "quadratic" parametrization
    aφ = ½ eiφ - 1/2 .
Therefore aφ lays on the r = 1/2 circle at the angle φ with respect to the real axis.

Period 2 orbit

The equation for the period 2 orbit zo = f o2(zo ) = f(f(zo )) is
    (zo2 + c)2 + c - zo = (zo2 + c - zo )(zo2 + zo + c + 1) = 0.
The roots of the first factor are the two fixed points z1,2 . They are repelling outside the main cardioid. The second factor has two roots
    z3,4 = -1/2 ± (-3/4 - c)½.
These two roots form period-2 orbit. Since z3 z4 = c + 1 the multiplier of the orbit is
    λ = f '(z3 ) f '(z4 ) = 4z3 z4 = 4(c + 1).
Therefore the orbit is attracting while |c + 1| < 1/4 or c lies within the ¼ exp(iφ) - 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). Two roots z3 , z4 are symmetrical with respect to the point z = -1/2.

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").

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updated 12 Sep 2013