4.4: Attracting Fixed point and Period 2 orbit

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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)

z_1,2 = 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 e^iφ ± 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

c_φ = ½ e^iφ - ¼ e^2iφ .

In the "quadratic" parametrization

a_φ = ½ e^iφ - 1/2 .

Therefore a_φ 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 z_o = f^o2(z_o) = f(f(z_o)) is

(z_o² + c)² + c - z_o = (z_o² + c - z_o)(z_o² + z_o + c + 1) = 0.

The roots of the first factor are the two fixed points z_1,2. They are repelling outside the main cardioid. The second factor has two roots

z_3,4 = -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 z_1-4 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").