Fractals/Iterations in the complex plane/MandelbrotSetExteriorComplex potential

Complex potential on the parameter plane

The green procedure is named after the British mathematician George Green

Complex potential is a complex number, so it has 2 parts:
 * a real part = real potential = absolute value
 * an imaginary part = external angle

One can take also its:
 * curl
 * divergence

So on one image one can use more than one variable to color image.

Implementations:
 * Matemathica

Names:
 * Boettcher coordinate

Real potential = CPM/M
Names:
 * eponimes:
 * Douady-Hubbard Potential
 * Hubbard-Douady potential
 * electric:
 * The electric potential
 * the voltage
 * Green function = G(c)

$$ G(c) = \lim_{n \to \infty} \frac{1}{2^n} ln|z_n|$$

$$V(c) \approx V_n(c) = \frac{log|z_n|}{2^n}$$

In Fractint : potential = log(modulus)/2^iterations

One can use real potential to:
 * smooth (continuous) coloring
 * discrete coloring ( level sets of potential)
 * 3D view

Code:
 * c code from mandel by Wolf Jung

Here is Delphi function which gives level of potential :

External angle and external ( parameter) ray
Example code ( Mathematica)

conjugate ange
Conjugate angle
 * angles of the wake

For the Mandelbrot set, considering strictly periodic rays with period >= 2, the other in the pair can be found in O(period^2) time using Henk Bruin and Dierk Schleicher's Symbolic Dynamics of Quadratic Polynomials : Algorithm 13.3:

Code:
 * https://code.mathr.co.uk/mandelbrot-symbolics/blob/82378e281c6ef149c7280d2b10076fffd83e0a8c:/c/lib/m_q_unlinked.c
 * https://code.mathr.co.uk/mandelbrot-symbolics/blob/82378e281c6ef149c7280d2b10076fffd83e0a8c:/c/lib/m_q_conjugate.c

Methods
First find angle of last iteration. It is easy to compute and shows some external rays as a borders of level sets. Then one go further.
 * binary decomposition of LSM/M
 * nth-decomposition : color of exterior is proportional to quadrant in which last iteration lands. ( see below )
 * exterior grid coordinates: (arg(final z),log(|final z|)/log(escaperadius))
 * stripe average coloring
 * automatically calculate external angles from nucleus and period by Claude Heiland-Allen

Methods:
 * newton method
 * Tomoki Kawahira paper
 * Wolf Jung explanation
 * speeding up external ray tracing by Mandelbrot set Newton basins - Claude Heiland-Allen
 * lines perpendicular to equipotential lines
 * field line calculations, images and videos by Michael T Everest
 * field lines by Chris M. Thomasson
 * using Runge-Kutta integration by
 * images
 * mandelbrot-phase-angle-gif-loop

The Wolf Jung test
The external parameter rays for angles (in turns)
 * 321685687669320/2251799813685247 (period 51, lands on c1 = -0.088891642419446 +0.650955631292636i )
 * 321685687669322/2251799813685247 ( period 51 lands on c2 = -0.090588078906990 +0.655983860334813i )
 * 1/7 ( period 3, lands on c3 = -0.125000000000000  +0.649519052838329i  )

Angles differ by about $$10^{-15}$$, but the landing points of the corresponding parameter rays are about 0.035 apart. It can be computed with Maxima CAS : (%i1) c1: -0.088891642419446 +0.650955631292636*%i; (%o1) 0.650955631292636*%i−0.088891642419446 (%i2) c2:-0.090588078906990 +0.655983860334813*%i; (%o2) 0.655983860334813*%i−0.09058807890699 (%i3) abs(c2-c1); (%o3) .005306692383854863 (%i4) c3: -0.125000000000000 +0.649519052838329*%i$ (%i5) abs(c3-c1); (%o5) .03613692356607755 (%i6) a3:1/7$ (%i7) float(abs(a3-a1)); (%o7) 4.440892098500628*10^−16 Informations from W Jung program : The angle 1/7  or  p001 has  preperiod = 0  and  period = 3. The conjugate angle is 2/7  or  p010. The kneading sequence is AA*  and the internal address is  1-3. The corresponding parameter rays are landing at the root of a satellite component of period 3. It is bifurcating from period 1.

The angle 321685687669320/2251799813685247  or  p001001001001001001001001001001001001001001001001000 has  preperiod = 0  and  period = 51. The conjugate angle is 321685687669319/2251799813685247  or  p001001001001001001001001001001001001001001001000111. The kneading sequence is AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABB*  and the internal address is  1-49-50-51. The corresponding parameter rays are landing at the root of a primitive component of period 51.

The angle 321685687669322/2251799813685247  or  p001001001001001001001001001001001001001001001001010 has  preperiod = 0  and  period = 51. The conjugate angle is 321685687669329/2251799813685247  or  p001001001001001001001001001001001001001001001010001. The kneading sequence is AABAABAABAABAABAABAABAABAABAABAABAABAABAABAABAABAA*  and the internal address is  1-3-51. The corresponding parameter rays are landing at the root of a satellite component of period 51. It is bifurcating from period 3.

The test by G. Pastor and Miguel Romera
The external parameter rays for angles (in turns) the central babies Mandelbrot sets of the cauliflowers located at -0.153756141 + 1.030383223i
 * 6871947673/34359738367 ( period 35 )
 * 9162596898/34359738367 ( period 35 )

(not that 34359738367 = 2^35 - 1)

test by M. Romera,1 G. Pastor, A. B. Orue,1 A. Martin, M.-F. Danca,and F. Montoya
G Pastor gave an example of external rays for which the resolution of the IEEE 754 is not sufficient:
 * $$\theta_{267}^- = 0.((001)^{88}010)_2 = \frac{33877456965431938318210482471113262183356704085033125021829876006886584214655562}{237142198758023568227473377297792835283496928595231875152809132048206089502588927}$$
 * $$\theta_{267}^+ = 0.((001)^{87}010001)_2 = \frac{33877456965431938318210482471113262183356704085033125021829876006886584214655569}{237142198758023568227473377297792835283496928595231875152809132048206089502588927}$$
 * $$\theta_{3}^- = 0.(001)_2 = \frac{1}{7} =  0.(142857)_{10}$$   ( period 3, lands on root point of period 3 component c3 = -0.125000000000000 +0.649519052838329i )
 * $$\theta_{268}^- = 0.((001)^{88}0001)_2 = \frac{67754913930863876636420964942226524366713408170066250043659752013773168429311121}{474284397516047136454946754595585670566993857190463750305618264096412179005177855}$$
 * $$\theta_{268}^+ = 0.((001)^{88}0010)_2 = \frac{67754913930863876636420964942226524366713408170066250043659752013773168429311122}{474284397516047136454946754595585670566993857190463750305618264096412179005177855}$$

One can analyze these angles using program by Claude Heiland-Allen :

Landing points of above rays are roots with angled internal addresses ( description by Claude Heiland-Allen) :
 * the upper one will be 1 -> 1/3 -> 3 -> 1/(period/3) -> period because it's the nearest bulb to the lower root cusp of 1/3 bulb and child bulbs of 1/3 bulb have periods 3 * denominator(internal angle) ie, 1 -> 1/3 -> 3 -> 1/89 -> 267
 * the lower one will be 1 -> floor(period/3)/period -> period because it's the nearest bulb below the 1/3 cusp ie, 1 -> 89/268 -> 268
 * the middle ray .(001) lands at the root of 1 -> 1/3 -> 3, from the cusp on the lower side (which is on the right in a standard unrotated view)

=References=