# 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

## Relation between external angles and Misurewicz point

The dynamics of the external angle under angle doubling is not the same as the dynamics of the landing point under iteration: the pre-period and period can differ between the external angle and the landing Misiurewicz point

Examples:

• the external angle 0.1(0) has pre-period 1 and period 1, and it lands on the point c = −2 = $M_{2,1}$  ( pre-period 2 and period 1)
• external angle .001001(010010100) ray is landing on the point c = 0.026593792304386393+0.8095285579867694i. The external angle have pre-period 6 and period 9, but landing point has pre-period 7 and period 3

# notation

Misiurewicz point (parameter, polynomial, map) can be marked by:

• preperiod and period
• the parameter coordinate c ∈ M
• the angle $\theta$  of the external ray

## angle

"Critically preperiodic polynomials are typically parameterized by the angle θ of the external ray landing at the critical value rather than by the critical value." MARY WILKERSON

The preperiodic angle $\theta$  (decimal fraction with even denominator ) of the external ray that lands:

• at z = c in the Julia set J(f) on the dynamic plane
• at $c=\gamma _{M}(q/p)=M_{t,p}$  in Mandelbrot set M on the parameter plane

so

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

Examples:

• Ray for angle ${\frac {1}{2}}=0.0(1)$  lands on the point $c=\gamma _{M}(1/2)=-2=M_{1,1}$  from the parameter plane. It is tip of the main antenna ( end of 1/2 limb). On the the dynamic plane it gives the Line Julia set
• Ray for angle ${\frac {1}{4}}=0.00(1)$  lands on the point $c=\gamma _{M}(1/4)=-0.228155493653962+1.115142508039937i=M_{2,1}$  from the parameter plane. It is the first tip of wake 1/3. On the the dynamic plane it gives the dendrite Julia set
• Ray for angle ${\frac {1}{6}}=0.0(01)$  lands on the point $c=\gamma _{M}(1/6)=i=M_{1,2}$  from the parameter plane. It is last tip of wake 1/3. On the the dynamic plane it gives the classic Dendrite Julia set

partition of the dynamic plane by dynamic rays related with the kneading sequence

## preperiod and period

$M_{t,p}$ where

• t is preperiod
• p is period

Preperiod is used in 2 meanings :

• T =preperiod of critical point $z_{cr}=0$
• t = preperiod of critical value $z_{cv}=c$

Note that :

$t=T-1$ Period p is the same for critical value and critical 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 $M_{2,1}$  is given as $M_{3,1}$  "

# 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

Visual types:

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

### spirals

The Misiurewicz points which are centers of the spirals can be classified according to speed of turning:

• fast
• slow
• no turn : if the Misiurewicz point is a real number, it does not turn at all

Spirals can also be classified by the number of arms.

Each Misiurewicz point has a multiplier which is related to the shape of the 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

The structural Misiurewicz points in an embedded Julia set all have the same period, that of the influencing island.

Tips:

• "It seems the tips of the nth-longest arms of primary spirals have period n, counting from 1, and the tips of some filaments have period 1" Claude Heiland-Allen
• not all terminal points have preperiod 1 or period 1

## 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

#### separators

##### primary separator
• band-merging point = point merging 2 chaotic bands $B_{i-1}$  and $B_{i}$
 $m_{i}=M_{2^{i},2^{i-1}}$ Examples: 

• $c=m_{0}=M_{1,1}\ \ =-2$  = tip of main antenna , external angle = 1/2
• $c=m_{1}=M_{2,1}\ \ =-1.543689012692076...$ , external angles 5 and 7/12
• $c=m_{2}=M_{4,2}\ \ =-1.430357632451307...$ , external angles 33 and 47/80
• $c=m_{3}=M_{8,4}\ \ =-1.407405118164702...$ , external angles 1795 and 2557/4352
• $c=m_{4}=M_{16,8}\ =-1.402492176358564...$
• $c=m_{5}=M_{32,16}=-1.401441494253588...$
• $c=m_{6}=M_{64,32}=-1.401216504309415..$
• $c=m_{7}=M_{128,64}=-1.401168320839301..$
• $c=m_{8}=M_{256,128}=-1.401158001505211..$
• $c=m_{9}=M_{512,256}=-1.401155791424613..$
• $c=m_{10}=M_{1024,512}=-1.401155318093230..$
• $c=m_{11}=M_{2048,1024}=-1.401155216720152..$
• ...
• $c=m_{\infty }=MF=-1.4011551890......$  = Feigenbaum point = MF = the Myrberg-Feigenbaum

In the text form:

double m = {
-2.0,
-1.543689012692076,
-1.430357632451307,
-1.407405118164702,
-1.402492176358564,
-1.401441494253588,
-1.401216504309415,
-1.401168320839301,
-1.401158001505211,
-1.401155791424613,
-1.401155318093230,
-1.401155216720152

};

const complex double cf = -1.401155189093314712; //the Feigenbaum point -1.401155 = m[infinity]

##### secondary separator
• tree separators ( tree is a subset of band )
 $m_{i,j}=M_{(j+1)2^{i}-1,2^{i}}$ ### non-characteristic

• have not the same period as the band

# Examples

Misiurewicz Points, part of the Mandelbrot set:

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

## wakes

### wake 1/2

  ${\begin{cases}0.(s_{-})=0.(01)={\frac {1}{3}}={\frac {4}{12}}=0.(3)=wake\\0.s_{-}(s_{+})=0.01(10)={\frac {5}{12}}=0.41(6)=PrincipalMis=M_{2,2}\\0.0(1)={\frac {1}{2}}={\frac {6}{12}}=0.5=tip=M_{1,1}=c=-2\\0.s_{+}(s_{-})=0.10(01)={\frac {7}{12}}=0.58(3)=PrincipalMis=M_{2,2}\\0.(s_{+})=0.(10)={\frac {2}{3}}={\frac {8}{12}}=0.(6)=wake\\\end{cases}}$ Important points of the wake:

• bond point = root point between period 1 and 2 components = c = -0.75 = -3/4 = birurcation point for internal angle 1/2 = Landing point of 2 external rays 1/3 and 2/3 = start of wake 1/2
• nucleus (center of component ) for period 2 = c = -1
• tip of main antenna c = -2 = $M_{1,1}$ . It is landing point of externa ray for angle $0.01={\frac {1}{2}}=0.5$
• c = -1.543689012692076 = Principal Misiurewicz point of wake 1/2 = main node ( branch point) of the wake = $M_{2,2}$  = landing point of external rays 5/12 i 7/12

## How to colour ?

"The legendary colour palette technique embeds an image in the iteration bands of an escape time fractal by linearizing it by scanlines and synchronizing the scan rate to the iterations in the fractal spirals so they line up to reconstruct the original image. Historically this has been done by preparing palettes for fractal software using external tools, and mostly only for small images (KF for example has a palette limited to 1024 colour slots).
Kalles Fraktaler 2 has an image texture feature, which historically only allowed you to warp a background through the semi-transparent fractal. I added the ability to create custom colouring algorithms in OpenGL shader language  (GLSL), with which it is possible to repurpose this texture and (for example) use it as a legendary palette.
Here I scaled my avatar (originally 256x256) to 128x16 pixels, and fine tuned the iteration count divisor by hand after zooming to a spiral in the Seahorse Valley of the Mandelbrot set. Then the face from the icon is visible in  the spirals all the way down to the end of the video. I used a work-in-progress (not yet released) build of KF 2.15.3, which has a new setting not to resize the texture to match the frame size: this allows the legendary technique  to work much more straightforwardly.
I rendered exponential map EXR frames from KF and assembled into a zoom video with zoomasm. From KF I exported just the RGB channels with the legendary palette colouring, and the distance estimate channels. I did not colour the  RGB with the distance estimate in KF, because with the exponential map transformation they would not be screen-space correct (the details would be smaller in the center of the reprojected video than at the edges). I could not do  all the colouring in zoomasm either, because it does not support image textures. I added the boundary of the fractal in zoomasm afterwards, by mixing pink with the RGB from KF according to the length of the screen-space distance  estimate channels (which zoomasm scales properly when reprojecting the exponential map)." Claude Heiland-Allen

# How to compute ...?

## Number of Misiurewicz points

• math.stackexchange question: counting-misiurewicz-points
• enumeration of misiurewicz points by Claude Heiland-Allen
• "... 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 
• The On-Line Encyclopedia of Integer Sequences (OEIS)
• $M_{0,p}$  is known to be A000740
• $M_{2,p}$  appears to be A038199
• $M_{q,1}$  appears to be A000225
• $M_{q,2}$  appears to be A166920
• Corollary 3.3. from Misiurewicz Points for Polynomial Maps and Transversality by Benjamin Hutz, Adam Towsley

The number of $(m,n)$  Misiurewicz points for $f_{d,c}$  is $M_{m,n}$ :

$M_{m,n}={\begin{cases}\sum _{k|n}\mu ({\frac {n}{k}})d^{k-1}&m=0\\(d^{m}-d^{m-1}-d+1)\sum _{k\mid n}\mu ({\frac {n}{k}})d^{k-1}&m\neq 0{\text{ and }}n\mid (m-1)\\(d^{m}-d^{m-1})\sum _{k\mid n}\mu ({\frac {n}{k}})d^{k-1}&{\text{otherwise}}\end{cases}}$

Where:

• m is
• n is
• $f_{d,c}(z)=z^{d}+c$
• d is a degree of $f$  function
• $\mu (n)$  is the Moebius function for the natural n
• $\sum _{k|n}\;\mu (r)$  is the sum of $\mu (r)$  over all positive integers $r$  dividing $n$

Implementations

## Misiurewicz points of complex quadratic mapping

• numerical methods
• solving equations
• finding landing point of external rays with pre-periodic angles
• graphical method
• Misiurewicz domains

"the best way of being sure you get to the correct point is to trace an external ray with the correct external angle, until you reach close enough (for example, the ray point has a tiny imaginary part, as these points are all on the real axis).  then use Newton's method starting from the ray point." Claude Heiland-Allen

### tracing rays

• "tracing rays of preperiod + period ~= 500 to dwell ~1000, with all 4 methods and varying sharpness" 
• "I made the database by tracing every pre-periodic ray with pre-period and period summing to less than or equal to 16. I traced the rays to the limit of double precision, averaging 400 rays per second on my quad core desktop. Then I grouped together the rays landing at the same point (or nearby, with a small threshold radius). I only grouped together rays having the same period and pre-period..."

### Solving equations

#### 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

## external angles of rays that land on the Misiurewicz point

Method:

• the external angle 0.1(0) has pre-period 1 and period 1, and it lands on the point c = −2 = $M_{2,1}$  ( pre-period 2 and period 1)