Periodic points in the Mandelbrot set
g3(c) = fco3(0) =
(c2 + c)2 + c =
c (c3 + 2c2 + c + 1) = 0
A point c in the Mandelbrot set is periodic point
with period n if its critical orbit is periodic with period n,
i.e. gn(c) = fcon(0) = 0. To the left
you see periodic critical orbit with period 3. As since
(f on)'(0) = f '(0) f '(z1) ...
f '(zn-1) =
2n 0 z1 ... zn-1 = 0 ,
therefore this orbit is superstable. For example:
g1(c) = fc(0) = c = 0
c1 [1 on the picture below] = 0
g2(c) = fco2(0) =
c2 + c = c (c + 1) = 0
c1 = 0, c2  = -1
c1 = 0, c2 [3a] = -1.75488,
c3 [3b] = c4* = -0.122561 + 0.744862i
g4(c) = fco4(0) = 0
c1 = 0, c2 = -1,
c3 [4a] = -1.9408, c4 [4b] = -1.3107,
c5 [4c] = c6* = -0.15652 + 1.03225i,
c7 [4d] = c8* = 0.282271 + 0.530061i
The number of such points doubles for each successive value of n
because gn(c) is a polynomial in c
of degree 2(n-1). It is known that it always has
2(n-1) distinct roots. If cn is a periodic
point, then cn* is periodic too.
It is evident, that every M-set bulb contains periodic point and corresponding
J-set has superstable period-n critical orbit. This point is the
nearest to the bulb "center" root of fc on(0) = 0
and it can be found e.g. by the Newton algorithm.
Preperiodic (Misiurewicz) points in the
A point Mk,n in M is preperiodic with period
n if its critical orbit becomes periodic with period n after
k (a finite number) steps.
It is evident, that preperiodic points Mk,n are
roots of equation:
fc ok(0) =
fc o(k+n)(0) or
gk(c) = gk+n(c) .
We have seen before that for given c the fixed points
z1 = 1/2 +- (1/4 - c)1/2 (i.e. period-1 orbits)
have multipliers l1 = 2z1
= 1 +- (1-4c)1/2. Therefore any preperiodic point
Mk,1 with period 1 has multiplier
1/2 - (1/4 - Mk,1)1/2 .
The plus sign corresponds to the only preperiodic point M2,1
= -2 (the tip of the Mandelbrot set antenna or the crisis point) with
the multiplier l = 4 . As since multiplier of
period-2 orbit is l2 = 4(c + 1)
therefore multiplier of period-2 Misiurewicz point Mk,2 is
4(Mk,2 + 1) .
Two examples are M2,1 = -2 and
M2,2 = i. Its critical orbits are
(0, -2, 2, 2,...) and
(0, i, i-1, -i, i-1, -i...)
respectively and their periods are 1 and 2.
The orbits are repelling. To see this, the relevant multipliers are
l1(-2) = 4
and l2(i) = 4(1 + i)
and all of these have absolute value exceeding 1.
Preperiodic points are not in a black region of M because there are
points arbitrarily close that do not belong to M.
Here are some preperiodic points with period 1.
All these points lie outside the main cardioid and the relevant fixed points
|3 ||3 || -0.22816+1.11514i
|| 3.08738 || -23.126|
Real periodic and preperiodic points
For real c real polynomials g1,2,...,5(c) are shown
in Fig.1. Real periodic points are roots of these polynomials.
An intersection of two curves gk(c) = gk+n(c)
corresponds to a Mk,n preperiodic point (M2,1
and M3,1 are shown here). Thus Fig.1 let us classify all
periodic and preperiodic points in a simple visual way (at least for small
Previous: Birth of Douady's rabbit
Next: Misiurewicz points and the M-set self-similarity
updated 19 August 2002