Fractals/Iterations in the complex plane/construct

How to construct a map with desired properities ?

Shape of

• Julia set
• critical orbit
• external rays landind on the repelling cycle ( spirals)

Relation between shape types and dynamics:

• n-th arm spiral: attracting or repelling n-periodic orbit ( cycle)
• closed curve: Siegel disc ( rotation)
• n-th arm star = period n parabolic root

Usually one should controll 2 parameters:

• fixed point
• period p orbit

• 
  Julia set is modelled by critical orbit
• 
  Julia set is modelled by critical orbit
• 
  Siegel disk critical orbit is modelled by period 7 repelling orbit
• 
  Siegel disk critical orbit is modelled by period 2 repelling orbit
• 
  Julia set is modelled by repelling period 6 orbit and it's external rays

See

• Shaping spirals

• 
  from Siegel to parabolic star
• 
  Superattracting through attracting spiral to parabolic star ( along internal ray)

• Rational functions with prescribed critical points by I. Scherbak
• constructing map with Fatou components of desired type^[1]
• Constructing polynomials whose Julia set resemble a desired shape^[2]
• Constructing polynomials that have an attracting cycle visiting predefined points in the plane ^[3]
• constructuing QUARTIC JULIA SETS INCLUDING ANY TWO COPIES OF QUADRATIC JULIA SETS
  • QUARTIC JULIA SETS INCLUDING ANY TWO COPIES OF QUADRATIC JULIA SETS by Koh Katagata
  • codepen
  • SE
• constructing critically finite real polynomial maps with specified combinatorics^[4]

The fundamental theorem of algebra states that every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots^[5][6]

The factor theorem^[7] states that a polynomial ${\displaystyle f(z)}$ has a factor ${\displaystyle (z-r)^{m}}$ if and only if ${\displaystyle f(r)=0}$ (i.e. ${\displaystyle r}$ is a root of multiplicity^[8] m)

Examples

The polynomial^[9]

 ${\displaystyle f(z)=(z-0)^{2}(z-1)^{2}(z+1)=z^{5}-z^{4}-z^{3}+z^{2}}$
 
 

The polynomial

 ${\displaystyle f(x)=(x+4)(x-1)^{2}=x^{3}+2x^{2}-7x+4}$
 

has roots:

• 1 of multiplicity 2
• -4 of multiplicity 1

p(z) = z^4 + O(z^2), where the four roots of p(z) are:

• one fixed at the origin,
• the remaining three forming the vertices of an equilateral triangle centered at the origin and rotating.

create polynomial with desired properities

• • f(z) = z*g(z) with root at origin
  • g(z) is a 3-rd root of unity = ${\displaystyle x^{2}+x+1}$

f(z) = z(z^2+z+1)

One can check it with Maxima CAS

(%i1) solve([z*(z^2+z+1)=0],[z]);
                      sqrt(3) %i + 1      sqrt(3) %i - 1
(%o1)          [z = - --------------, z = --------------, z = 0]
                            2                   2
(%i2) 

to rotate it around origin let's change 1 with :${\displaystyle \quad e^{2\pi it}}$ wher t is a proper fraction in turns

${\displaystyle f_{t}(z)=z(z^{2}+z+e^{2\pi it})}$

See : Vector_field short videos by boyce1000

System of 2 equations:

${\displaystyle {\begin{cases}f_{c}^{p}(z_{p})=z\\{\frac {d}{dz}}f_{c}^{p}(z_{p})=r*e^{2\pi \theta i}\end{cases}}}$

where:

• ${\displaystyle f_{c}}$ is a rational function with one parameter c
• ${\displaystyle \ f_{c}^{(p)}(z)}$ is the ${\displaystyle p}$-fold composition of ${\displaystyle f_{c}\,}$ with itself
• ${\displaystyle z_{p}}$ is a cyclic point ( point of limit cycle)
• p is a period of the cycle
• ${\displaystyle \lambda =r*e^{2\pi \theta i}}$ is a multiplier^[10] ( complex number)
• ${\displaystyle r=|\lambda |}$ is a stability of the cycle ( Real number )

Input :

• function ${\displaystyle f}$
• p ( integer)
• r ( real number)
• ${\displaystyle \theta }$ ( real number or rational number)

Unknowns ( solutions or output):

• parameter c ( complex number)
• periodic point ${\displaystyle z_{p}}$ ( complex number)

Maxima CAS program:

/*

batch("m.mac");

*/

display2d:false$
kill(all)$
ratprint:false$

/* complex quadratic polynomial */
f(z,c):= z*z+c $

/* iterated function */
F(z, c, n) :=
       if n=1 then f(z,c)
        else f(F(z, c, n-1),c)$
        
        
        
/*  multiplier = first deric=vative */        
m(z,c,p):= diff(F(z,c,p),z,1)$

l(r,t) := float(rectform(r*exp(2*%pi*t*%i)))$

/* input */

p:5$
r:1.0$
t:0$

/* system of equations */
e1: F(z,c,p)=z;
e2: m(z,c,p)=l(r,t);

/* 
output = solutions = 2 complex number: c, z 
*/

s:solve([e1,e2])$
s:map('float,s)$
s:map('rectform,s);

Example output:

For :

• p = 3
• r=1.0
• t=0.0

[
[z = 0.5,c = 0.25],
[z = (-0.4330127018922193*%i)-0.25,c = (-0.6495190528383289*%i)-0.125],
[z = 0.4330127018922193*%i-0.25,c = 0.6495190528383289*%i-0.125],
[z = -0.05495813133539004,c = -1.75],
[z = 1.301937809824245,c = -1.75],
[z = -1.746979634104245,c = -1.75]
]

For :

• p = 5
• r=1.0
• t=0.0

[
[z = 0.5,c = 0.25],
[z = 0.4755282581475767*%i+0.1545084971874737,c = 0.3285819450744551*%i+0.3567627457812099],
[z = 0.1545084971874737-0.4755282581475767*%i, c = 0.3567627457812106-0.3285819450744586*%i],
[z = 0.2938926261462365*%i-0.4045084971874737, c = 0.5316567552200239*%i-0.4817627457812153],
[z = (-0.2938926261462365*%i)-0.4045084971874737, c = (-0.5316567552199369*%i)-0.481762745781224],
[z = -0.003102011282477321,c = -1.985409652076318],
[z = 0.0109289978340113,c = -1.860587002096436],
[z = 8.008393221517376E-4*%i-0.01213161194929343, c = 1.100298437397382*%i-0.1978729466687337],
[z = (-8.008393221517376E-4*%i)-0.01213161194929343, c = (-1.100298437397305*%i)-0.1978729466687667],
        [z = 0.02151217276434695*%i-0.005267866463337371, c = 0.3797412022535638*%i-1.256801993945385],
        [z = (-0.02151217276434695*%i)-0.005267866463337371, c = (-0.3797412022517599*%i)-1.256801993944077],
        [z = 0.02591758988716001*%i+0.0096648625988135, c = 0.9868115621249533*%i-0.04506136597934137],
        [z = 0.0096648625988135-0.02591758988716001*%i, c = (-0.9868115621250132*%i)-0.04506136597930513],
        [z = -0.02506558296814108,c = -1.624396967608546],
        [z = 0.02532354987824971*%i-0.0286751769590709, c = 0.6415066667139064*%i+0.3599331333357185],
        [z = (-0.02532354987824971*%i)-0.0286751769590709,
         c = 0.3599331333357186-0.6415066667139071*%i], [z = 0.7018214526647177,c = -1.860587002096436],
        [z = 0.5745382937725365*%i+0.1798116252110209, c = (-0.379741202251533*%i)-1.25680199394442],
        [z = 0.1798116252110209-0.5745382937725365*%i, c = 0.3797412022514344*%i-1.256801993944486],
        [z = -0.5997918293000261,c = -1.624396967608546],
        [z = 0.6400543521659254*%i+0.3601141169309163, c = 0.6415066667138928*%i+0.3599331333356947],
        [z = 0.3601141169309163-0.6400543521659254*%i, c = 0.3599331333356951-0.6415066667138929*%i],
        [z = 0.747361547631752*%i+0.4122389750905872, c = 0.3599331333377524-0.6415066667118048*%i],
        [z = 0.4122389750905872-0.747361547631752*%i,c = 0.6415066667118131*%i+0.3599331333377574],
        [z = -1.264646754738656,c = -1.624396967608546],
        [z = 0.838427461519175*%i+0.1867295812979602,c = (-0.9868115621248*%i)-0.04506136597962632],
        [z = 0.1867295812979602-0.838427461519175*%i, c = 0.9868115621248269*%i-0.04506136597961512],
        [z = 1.012227741688957,c = -1.624396967608546],
        [z = 0.6736931444481549*%i-0.7131540376767388,  c = 0.9868115621009495*%i-0.04506136566593825],
        [z = (-0.6736931444481549*%i)-0.7131540376767388, c = (-0.9868115621015654*%i)-0.04506136566602404],
        [z = 0.6816651712455555*%i+0.8064792250322852,  c = (-1.100298438532418*%i)-0.1978729463920518],
        [z = 0.8064792250322852-0.6816651712455555*%i,c = 1.100298438531886*%i-0.197872946387467],
        [z = 0.9873125420152975*%i-0.04563967787575593, c = 0.9868115621249436*%i-0.04506136597927069],
        [z = (-0.9873125420152975*%i)-0.04563967787575593, c = (-0.9868115621249249*%i)-0.04506136597929692],
        [z = -1.368033648790746,c = -1.860587002096436],
        [z = -1.623768668573244,c = -1.624396967608546],
        [z = 1.600752508361204,c = -1.860587002096436],
        [z = 0.8177857184842046*%i-0.8491638964763748,  c = 0.6415066726649287*%i+0.3599331357137042],
        [z = (-0.8177857184842046*%i)-0.8491638964763748, c = 0.3599331357115682-0.6415066726792946*%i],
        [z = -1.860467532467532,c = -1.860586580956207], 
        [z = 0.1585230889211015*%i+1.129895436404861,  c = (-0.3797412017812437*%i)-1.256801993890818],
        [z = 1.129895436404861-0.1585230889211015*%i,  c = 0.3797412020742688*%i-1.256801993924219],
        [z = 1.102491882350288*%i+0.07994573682221373, c = 0.641506666713125*%i+0.3599331333375105],
        [z = 0.07994573682221373-1.102491882350288*%i, c = 0.3599331333375118-0.641506666713142*%i],
        [z = 1.10027900645412*%i-0.1977264120044163,c = 1.100298437399976*%i-0.1978729466589521],
        [z = (-1.10027900645412*%i)-0.1977264120044163, c = (-1.100298437392994*%i)-0.1978729466579122],
        [z = 0.3795145554958574*%i-1.257237017109811,  c = 0.3797412012322979*%i-1.256801993538778],
        [z = (-0.3795145554958574*%i)-1.257237017109811, c = (-0.3797412011893692*%i)-1.256801993401957],
        [z = 0.8966903093631682*%i-1.01776444141452, c = 0.986811439368143*%i-0.04506141337632084],
        [z = (-0.8966903093631682*%i)-1.01776444141452, c = (-0.9868114393633113*%i)-0.04506141338736716],
        [z = 1.407944514501891,c = -1.985409652076318],
        [z = 0.7215120925377011*%i+1.234881318742427,  c = (-1.100298500720014*%i)-0.1978727350763138],
        [z = 1.234881318742427-0.7215120925377011*%i,c = 1.100298500782114*%i-0.1978727352231734],
        [z = 0.6651899971189704*%i-1.369391104706556, c = 1.100298438532065*%i-0.1978727774731155],
        [z = (-0.6651899971189704*%i)-1.369391104706556, c = (-1.100298478086625*%i)-0.1978727911942495],
        [z = 0.1731238730127708*%i-1.554564024233688,  c = 0.3797412149717089*%i-1.256801976456581],
        [z = (-0.1731238730127708*%i)-1.554564024233688, c = (-0.3797411926534995*%i)-1.256801968631482],
        [z = 1.842105908761944,c = -1.985410334346504],
        [z = 1.956403762662807,c = -1.985409652076318]
        ]

Mandelbrot Set - Convergent Evolution of P/Q Limbs in Seahorse Valley by izaytsev0

• main cardioid seahorse valley = Gap between the head ( period 2 component) and the body (or shoulder = main cardioid). Particularly the upper one part
• 2 windows
  • left: limbs from period 2 component
  • right: limbs from period 1 componnet
• in each window one can see limb p/q from 14/30 on the right and increasing p ?

Compare with Real dense fractal Zoom! Part 2 by SeryZone Arts

• perturbation
• math.stackexchange question: how-can-i-reconstruct-a-julia-set-by-a-given-image ?

1. ↑ fractalforums: julia-sets-true-shape-and-escape-time
2. ↑ fractalforums : constructing-polynomials-whose-julia-set-resemble-a-desired-shape
3. ↑ fractalforums : constructing-polynomials-with-attracting-cycles
4. ↑ The W. Thurston algorithm applied to real polynomial maps Araceli Bonifant, J. Milnor, S. Sutherland Published 15 May 2020
5. ↑ The fundamental theorem of algebra in wikipedia
6. ↑ Ed Pegg Jr "The Fundamental Theorem of Algebra" http://demonstrations.wolfram.com/TheFundamentalTheoremOfAlgebra/ Wolfram Demonstrations Project Published: November 10 2011
7. ↑ Factor theorem in wikipedia
8. ↑ Multiplicity_of_a_root_of_a_polynomial in wikipedia
9. ↑ fractalforums.org : julia-sets-true-shape-and-escape-time
10. ↑ Stability_of_periodic_points_(orbit)_-_multiplier in wikipedia

• Bishop, Christopher J. Constructing entire functions by quasiconformal folding. Acta Math. 214 (2015), no. 1, 1--60. doi:10.1007/s11511-015-0122-0.
• Constructing entire functions with non-locally connected Julia set by quasiconformal surgery by Yanhua Zhang, Gaofei Zhang
• Kumar, Rajen & Nayak, Tarakanta. (2020). The real non-attractive fixed point conjecture and beyond.
• Specifying attracting cycles for Newton maps of polynomials by James T. Campbell, Jared T. Collins
• Exploring the Mandelbrot set. The Orsay Notes. Adrien Douady John H. Hubbard: constructing a polynomial with a given tree
• Godillon, S'ebastien. “Construction of rational maps with prescribed dynamics.” (2010).
• On building polynomials, November 2005The Mathematical Gazette 89(516), DOI: 10.2307/3621936, Christopher Sangwin