# Fractals/Iterations in the complex plane/construct

How to construct a map with desired properities ?

## Shape

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

### Modelling or shaping

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

### Conversions and evolutions

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

## Examples

* Rational functions with prescribed critical points by I. Scherbak
* constructing map with Fatou components of desired type
* Constructing polynomials whose Julia set resemble a desired shape
* Constructing polynomials that have an attracting cycle visiting predefined points in the plane
* 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

### roots

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
The factor theorem states that a polynomial $f(z)$ has a factor $(z-r)^{m}$ if and only if $f(r)=0$ (i.e. $r$ is a root of multiplicity m)
Examples
The polynomial
The polynomial
has roots:
* 1 of multiplicity 2
* -4 of multiplicity 1

#### rboyce1000

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 = $x^{2}+x+1$
f(z) = z(z^2+z+1)
One can check it with Maxima CAS
to rotate it around origin let's change 1 with :$\quad e^{2\pi it}$ wher t is a proper fraction in turns
    $f_{t}(z)=z(z^{2}+z+e^{2\pi it})$
See : Vector_field short videos by boyce1000

### one parameter

System of 2 equations:
${\begin{cases}f_{c}^{p}(z_{p})=z\\{\frac {d}{dz}}f_{c}^{p}(z_{p})=r*e^{2\pi \theta i}\end{cases}}$
where:
* $f_{c}$ is a rational function with one parameter c
* $\ f_{c}^{(p)}(z)$ is the $p$-fold composition of $f_{c}\,$ with itself
* $z_{p}$ is a cyclic point ( point of limit cycle)
* p is a period of the cycle
* $\lambda =r*e^{2\pi \theta i}$ is a multiplier ( complex number)
* $r=|\lambda |$ is a stability of the cycle ( Real number )
Input :
* function $f$
* p ( integer)
* r ( real number)
* $\theta$ ( real number or rational number)
Unknowns ( solutions or output):
* parameter c ( complex number)
* periodic point $z_{p}$ ( complex number)
Maxima CAS program:
Example output:
For :
* p = 3
* r=1.0
* t=0.0
For :
* p = 5
* r=1.0
* t=0.0

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

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

## See also

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

## References

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