Attracting Fixed point and Period 2 orbit

The main cardioid equation

We have seen before that there are always two fixed points   z* = f(z*)   for a quadratic map
    f(z*) - z* = z*2 + c - z* = 0,     z1,2 = 1/2 -+ (1/4 - c)1/2.       (*)
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* .

The M-set in the "square" parametrization

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 .
That is the c = 1/4 - a2 transformation converts the main cardioid into the left circle.

Internal angles theory

Point on the main cardioid corresponding to an internal angle f = 2p m/n lies at
    cf = eif/2 - e2if/4 .

In "square" parametrization
    af = eif /2 - 1/2 .
Therfore af lays on the r = 1/2 circle at the angle f with respect to the real axis.

Period 2 orbit

Equation for 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)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). Two roots z3 , z4 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").


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updated 8 October 2000