# Fractals/Iterations in the complex plane/misiurewicz

Misiurewicz point is the parameter c ( point oc parameter plane) where the critical orbit is pre-periodic.

# Properities

## notation

Misiurewicz polynomial ( map) can be marked by:

• the parameter coordinate c ∈ M
• the external angle $\theta$ of the ray that lands:
• at z = c in J(f) on the dynamic plane
• at c in M on the parameter plane

$c=\gamma _{M}(p/q)$ so

$z^{2}+c=z^{2}+\gamma _{M}(p/q)$ Examples:

• the Kokopelli Julia set $c=\gamma _{M}(3/15)=0.156520166833755+1.032247108922832i$ The angle 3/15 = p0011 = 0.(0011) has preperiod = 0 and period = 4. The conjugate angle on the parameter plane is 4/15 or p0100. The kneading sequence is AAB* and the internal address is 1-3-4. The corresponding parameter rays are landing at the root of a primitive component of period 4.

## types

### period

Misiurewicz points c

• with period 1 are of the type:
• alpha, i.e. $f_{c}^{k}(c)=\alpha _{c}$ • beta, i.e $f_{c}^{k}(c)=\beta _{c}$ • with period > 1

### Topological

all Misiurewicz points are centers of the spirals, which are turning:

• slow
• fast
• if the Misiurewicz point is a real number, it does not turn at all

Spirals can also be classified by the number of arms.

Visual types:

• branch tips = terminal points of the branches or tips of the midgets
• centers of spirals = fast spiral
• branch point = points where branches meet = centers of slow spirals with more then 1 arm
• band-merging points of chaotic bands (the separator of the chaotic bands $B_{i-1}$ and $B_{i}$ ) = 2 arm spiral

### angles of external rays

• endpoint = 1 angle
• primitive type = 2 angles of primitive cycle
• satellite type = 2 or more angles from satellite cycle

### named types

#### principal

The principal Misiurewicz point $c=b$ of the limb $M_{k/m}$ :

• $f^{m}(b)=\alpha _{b}$ • hase m external angles, that are preimages (under doubling) of the external angles of $\alpha _{b}$ #### characteristic

Characteristic Misiurewicz point of the chaotic band of the Mandelbrot set is :

• the most prominent and visible Misiurewicz point of a chaotic band
• have the same period as the band
• have the same period as the gene of the band

# Examples

Misiurewicz Points, part of the Mandelbrot set:

• Centre 0.4244 + 0.200759i; Max. Iterations 100; View radius 0.00479616 

# Computing

"... we do not know how to compute (...) Misiurewicz parameters (with high (pre)periods) for the family of quadratic rational maps. One might need to and a non-rigorous method to and Misiurewicz parameter in a reasonable time like Biham-Wenzel's method." HIROYUKI INOU 

## Computing Misiurewicz points of complex quadratic mapping

### roots of polynomial

Misiurewicz points  are special boundary points.

Define polynomial in Maxima CAS :

P(n):=if n=0 then 0 else P(n-1)^2+c;


Define a Maxima CAS function whose roots are Misiurewicz points, and find them.

M(preperiod,period):=allroots(%i*P(preperiod+period)-%i*P(preperiod));


Examples of use :

(%i6) M(2,1);
(%o6) [c=-2.0,c=0.0]
(%i7) M(2,2);
(%o7) [c=-1.0*%i,c=%i,c=-2.0,c=-1.0,c=0.0]


### factorizing the polynomials

" factorizing the polynomials that determine Misiurewicz points. I believe that you should start with

  ( f^(p+k-1) (c) + f^(k-1) (c) ) / c


This should already have exact preperiod k , but the period is any divisor of p . So it should be factorized further for the periods.

Example: For preperiod k = 1 and period p = 2 we have

  c^3 + 2c^2 + c + 2 .


This is factorized as

(c + 2)*(c^2 + 1)


for periods 1 and 2 . I guess that these factors appear exactly once and that there are no other factors, but I do not know."Wolf Jung