# 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:[2]

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

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

so

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

Examples:

• the Kokopelli Julia set ${\displaystyle c=\gamma _{M}(3/15)=0.156520166833755+1.032247108922832i}$ [3] 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.

### preperiod

Preperiod is used in 2 meanings :

• K =preperiod of critical point
• k = preperiod of critical value

Note that :

 k = K -1


Period p is the same for critical value and citical point

Wolf Jung uses preperiod of critical value : "... the usual convention is to use the preperiod of the critical value. This has the advantage, that the angles of the critical value have the same preperiod under doubling as the point, and the same angles are found in the parameter plane."

Pastor uses preperiod of critical point : "all the Misiurewicz points are given with one unit more in their preperiods, therefore this M2,1 is given as M3,1 "[4]

## types

### period

Misiurewicz points c

• with period 1 are of the type:[5]
• alpha, i.e. ${\displaystyle f_{c}^{k}(c)=\alpha _{c}}$
• beta, i.e ${\displaystyle f_{c}^{k}(c)=\beta _{c}}$
• with period > 1

### Topological

all Misiurewicz points are centers of the spirals, which are turning:[6]

• 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:[7]

• branch tips = terminal points of the branches[8] or tips of the midgets[9]
• centers of spirals = fast spiral
• branch point = points where branches meet[10] = centers of slow spirals with more then 1 arm
• band-merging points of chaotic bands (the separator of the chaotic bands ${\displaystyle B_{i-1}}$ and ${\displaystyle B_{i}}$ )[11] = 2 arm spiral

### number of external rays

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

### preperiod and period

In general a preperiodic critical value has a preperiod k, a period p, a ray period rp, and v angles. There are three cases,

• tip: r = 1 and v = 1
• primitive: r = 1 and v = 2
• satellite: r > 1 and v = r

### named types

#### principal

The principal Misiurewicz point ${\displaystyle c=b}$ of the limb ${\displaystyle M_{k/m}}$:[12]

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

#### characteristic

Characteristic Misiurewicz point of the chaotic band of the Mandelbrot set is :[13]

• 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 [14]

# 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 [15]

## Computing Misiurewicz points of complex quadratic mapping

### roots of polynomial

Misiurewicz points [16] 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