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,
z1,2 = 1/2 ∓ (1/4 - c)½.
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∗ -
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
½ eiφ -
¼ e2iφ .
In the "quadratic" parametrization
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