Periodic points in the Mandelbrot set

A point c in the Mandelbrot set is periodic point
with period n if its critical orbit is periodic with period n,
i.e. g_{n}(c) = f_{c}^{on}(0) = 0. To the left
you see periodic critical orbit with period 3. As since
(f ^{on})'(0) = f '(0) f '(z_{1}) ...
f '(z_{n1}) =
2^{n} 0 z_{1} ... z_{n1} = 0 ,
therefore this orbit is superstable. For example:
g_{1}(c) = f_{c}(0) = c = 0
c_{1} [1 on the picture below] = 0
g_{2}(c) = f_{c}^{o2}(0) =
c^{2} + c = c (c + 1) = 0
c_{1} = 0, c_{2} [2] = 1

g_{3}(c) = f_{c}^{o3}(0) =
(c^{2} + c)^{2} + c =
c (c^{3} + 2c^{2} + c + 1) = 0
c_{1} = 0, c_{2} [3a] = 1.75488,
c_{3} [3b] = c_{4}^{*} = 0.122561 + 0.744862i
g_{4}(c) = f_{c}^{o4}(0) = 0
c_{1} = 0, c_{2} = 1,
c_{3} [4a] = 1.9408, c_{4} [4b] = 1.3107,
c_{5} [4c] = c_{6}^{*} = 0.15652 + 1.03225i,
c_{7} [4d] = c_{8}^{*} = 0.282271 + 0.530061i
The number of such points doubles for each successive value of n
because g_{n}(c) is a polynomial in c
of degree 2^{(n1)}. It is known that it always has
2^{(n1)} distinct roots. If c_{n} is a periodic
point, then c_{n}^{*} is periodic too.
It is evident, that every Mset bulb contains periodic point and corresponding
Jset has superstable periodn critical orbit. This point is the
nearest to the bulb "center" root of f_{c}^{ on}(0) = 0
and it can be found e.g. by the Newton algorithm.
Preperiodic (Misiurewicz) points in the
Mandelbrot set
A point M_{k,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 M_{k,n} are
roots of equation:
f_{c}^{ ok}(0) =
f_{c}^{ o(k+n)}(0) or
g_{k}(c) = g_{k+n}(c) .
We have seen before that for given c the fixed points
z_{1} = 1/2 + (1/4  c)^{1/2} (i.e. period1 orbits)
have multipliers λ_{1} = 2z_{1}
= 1 + (14c)^{1/2}. Therefore any preperiodic point
M_{k,1} with period 1 has multiplier
λ_{1} =
1/2  (1/4  M_{k,1})^{1/2} .
The plus sign corresponds to the only preperiodic point M_{2,1}
= 2 (the tip of the Mandelbrot set antenna or the crisis point) with
the multiplier λ = 4 . As since multiplier of
period2 orbit is λ_{2} = 4(c + 1)
therefore multiplier of period2 Misiurewicz point M_{k,2} is
λ_{2} = 4(M_{k,2} + 1) .
Two examples are M_{2,1} = 2 and
M_{2,2} = i. Its critical orbits are
(0, 2, 2, 2,...) and
(0, i, i1, i, i1, i...)
respectively and their periods are 1 and 2.
The orbits are repelling. To see this, the relevant multipliers are
λ_{1}(2) = 4
and λ_{2}(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
are repelling.
num  k  c  λ  Arg(λ)^{o} 
1  2  2  4  0 
2  3  1.54369  1.67857  180 
3  3  0.22816+1.11514i
 3.08738  23.126 
4  4  1.89291  1.92774  180 
5  4  1.29636+0.44185i  3.52939  5.7209 
6  4  0.10110+0.95629i  1.32833  119.553 
7  4  0.34391+0.70062i  2.45805  30.988 
Real periodic and preperiodic points

For real c real polynomials g_{1,2,...,5}(c) are shown
in Fig.1. Real periodic points are roots of these polynomials.
An intersection of two curves g_{k}(c) = g_{k+n}(c)
corresponds to a M_{k,n} preperiodic point (M_{2,1}
and M_{3,1} are shown here). Thus Fig.1 let us classify all
periodic and preperiodic points in a simple visual way (at least for small
n).

Contents
Previous: Birth of Douady's rabbit
Next: Misiurewicz points and the Mset selfsimilarity
updated 19 August 2002