# 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*.
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*^{(n-1)}. It is known that it always has
*2*^{(n-1)} distinct roots. If *c*_{n} is a periodic
point, then *c*_{n}^{*} 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 *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)^{½} (i.e. period-1 orbits)
have multipliers *λ*_{1} = 2z_{1}
= 1 ∓ (1-4c)^{½}. Therefore any preperiodic point
*M*_{k,1} with period 1 has multiplier

*λ*_{1} =
1/2 - (1/4 - M_{k,1 })^{½} .

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
period-2 orbit is *λ*_{2} = 4(c + 1)
therefore multiplier of period-2 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, 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

*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: Shaggy midgets
Next: Misiurewicz points and the M-set self-similarity

*updated* 12 Sep 2013