Fractals/Iterations in the complex plane/pcheckerboard

=Task= How to show dynamics inside parabolic basin ?

Parameter c is a parabolic point

Attractor (limit set, attracting cycle / fixed point )
 * belongs to Julia set ( not as in other cases )
 * is weakly attracting ( lazy dynamics)

=Algorithm=

Color depends on:
 * sign of imaginary part of numerically approximated Fatou coordinate (numerical explanation )
 * position of $$z_{\infty}$$ under $$f^p$$ in relation to one arm of critical orbit star: above or below ( geometric explanation)



Name of the algorithm

 * Parabolic chessboard = parabolic checkerboard
 * zeros of qn - use only interior of Filled Julia sets on real slice of Mandelbrot set ( c is a real number, iaginary part is zero)
 * Binary decomposition ( method = BDM ) in parabolic case
 * Tessellation of the Interior of Filled Julia Sets ( Tomoki Kawahira in Semiconjugacies in Complex Dynamics with Parabolics)
 * internal tiling

Steps
First choose:
 * choose target set, which is a circle with :
 * center in parabolic fixed point
 * radius such small that width of exterior between components is smaller then pixel width
 * Target set consist of fragments of p components ( sectors). Divide each part of target set into 2 subsectors ( above and below critical orbit ) = binary decomposition

Steps:
 * take $$z_0$$ = initial point of the orbit ( pixel)
 * make forward iterations
 * if points escapes = exterior
 * if point do not escapes then check if point is near fixed point ( in the target set)
 * if no then make some extra iterations
 * if is then check in what half of target set it ( binary decomposition)

How to find trap ?
Steps for parabolic basin
 * choose component with critical point inside
 * choose trap
 * decompose trap disk for binary decomposition = divide into 2 parts : above and beyond the critical orbit

Trap features:
 * is inside component with critical point inside
 * trap has parabolic point on it's boundary
 * center of the trap is  midpoint between last point of critical orbit and fixed point
 * radius of the trap is half of distance between fixed point and last point of critical orbit

How to compute preimages of critical point ?
Here a and b are 2 inverse ( or backward ) iterations ( multivalued with argument adjusted ) used in the program Mandel by Wolf Jung:
 * 1-st inverse function = key a = $$-f^{-1}(z) $$
 * 2-nd inverse function = key b = $$+f^{-1}(z) $$

"Use the keys a and b for the inverse mapping. (The two branches are approximately mapping to the parts A and B of the itinerary.)

=dictionary=
 * The chessboard is the name of this decomposition of A into a graph and boxes
 * the chessboard graph
 * the chessboard boxes :" The connected components of its complement in A are called the chessboard boxes (in an actual chessboard they are called squares but here they have infinitely many corners and not just four). "
 * the two principal or main chessboard boxes
 * trap = target set = attracting petal

Visualizing Structures with the Chessboard Graph

"An often used and very useful technique of visualization of ramified covers (and partial cover structures that are not too messy) consists in cutting the range in domains, often simply connected, along lines joining singular values, and taking the pre-image of these pieces, which gives a new set of pieces. The way they connect together and the way they map to the range give information about the structure." Arnaud Chéritat Near Parabolic Renormalization for Unicritical Holomorphic Maps by Arnaud Chéritat

=description=

"A nice way to visualize the extended Fatou coordinates is to make use of the parabolic graph and chessboard."

Color points according to :
 * the integer part of Fatou coordinate
 * the sign of imaginary part

Corners of the chessboard ( where four tiles meet ) are precritical points

$$\bigcup_{n=0}^{n \geq 0} f^{-n*p}(z_{cr})$$

or

$$\{ z: f^{n*p}(z) = z_{cr} \}$$

1/1
The parabolic chessboard for the polynomial z + z^2  normalizing  $$\psi_{att}(-1/2)= 0$$ * each yellow tile biholomorphically maps to the upper half plane * each blue tile biholomorphically maps to the lower half plane under $$\psi_{att}$$ * The pre-critical points of $$z + z^2$$ or equivalently the critical points of $$\psi_{att}$$ are located where four tiles meet"

=Images= Click on the images to see the code and descriptions on the Commons !

0/1


Cpp code by Wolf Jung see function parabolic from file mndlbrot.cpp ( program mandel )

To see effect :
 * run Mandel
 * (on parameter plane ) find parabolic point for angle 0, which is c=0.25. To do it use key c, in window input 0 and return.

C code :

Gnuplot code :

1/2 or fat basilica
Cpp code by Wolf Jung see function parabolic from file mndlbrot.cpp ( program mandel ) To see effect :
 * run Mandel
 * (on parameter plane ) find parabolic point for angle 1/2, which is c=-0.75. To do it use key c, in window input 0 and return.

C code :

1/3
Numerical approximation of Julia set for fc(z)= z^2 + c child_period = 3 internal argument in turns = 1 / 3 parameter c = -0.1250000000000000 +0.6495190528383290*I fixed point alfa z = a = -0.2500000000000000 +0.4330127018922194*I external angles of rays landing on the fixed point : t = 1/7 	t = 2/7 	t = 4/7 critical point z = zcr = 0.0000000000000000 +0.0000000000000000*I precritical point z = z_precritical = -0.2299551351162811 -0.1413579816050052*I external argument in turns of first ray landing on fixed point = 1 / 7

1/4
Julia set for fc(z)= z^2 + c internal argument in turns = 1/4 parameter c = 0.2500000000000000 +0.5000000000000000*I fixed point alfa z = a = 0.0000000000000000 +0.5000000000000000*I critical point z = zcr = 0.0000000000000000 +0.0000000000000000*I precritical point z = z_precritical = -0.2288905993372869 -0.0151096456992677*I external angles of rays landing on fixed point: 1/15, 2/15, 4/15, 8/15 ( in turns)

5/11
Numerical approximation of Julia set for fc(z)= z^2 + c parameter c = ( -0.6900598700150440 ; 0.2760264827846140 ) fixed point alfa z = a = ( -0.4797464868072486 ; 0.1408662784207147 ) external angle of ray landing on fixed point: 341/2047

=See also =
 * parabolic perturbation
 * Checkerboard in Hyperbolic tilings by User:Tamfang : images and Python code
 * https://plus.google.com/110803890168343196795/posts/Eun6pZVkkmA
 * shadertoy: Orbit trapped julia Created by maeln in 2016-Jan-19
 * Holomorphic checkerboard by etale_cohomology
 * Sepals of cauliflower
 * wikipedia :Zebra striping in computer graphics
 * aproximation cauliflower by the true tree: tree which converge to the cauliflower julia set. the set of vertices  is in one to one correspondence with  the grand orbit of the critical point of  z squared plus one quarter.

=references=