Fractals/Iterations in the complex plane/fprays

Parabolic fixed point of period p is a landing point of p dynamic external rays. These rays divide neighborhood into curvilinear sectors.

=On the main cardioid=

Maxima CAS code :

kill(all); remvalue(all);

DoublingMap(r):= block([d,n], n:ratnumer(r), d:ratdenom(r), mod(2*n,d)/d)$

GivePeriod (r):= block([rNew, rOld, period, pMax, p],     pMax:100,      period:0,      p:1,       rNew:DoublingMap(r),      while ((p<pMax) and notequal(rNew,r)) do        (rOld:rNew, rNew:DoublingMap(rOld), p:p+1 ),     if equal(rNew,r) then period:p,      period );

/* f(z) is used as a global function I do not know how to put it as a argument */

GiveOrbit(r0,OrbitLength):= block( [r,Orbit], r:r0, Orbit:[r], for i:1 thru OrbitLength step 1 do        ( r:DoublingMap(r), Orbit:endcons(r,Orbit)), return(sort(Orbit))

)$

compile(all);

R: 4985538889/17179869183; p: GivePeriod(R);

orbit:GiveOrbit(R, p);

/* angles around critical point */ e1:first(orbit); e2:last(orbit);

1/p
In the elephant valley ( from parameter plane ) it is easy to find rays landing on the roots and dynamic external rays that land on the parabolic fixed point z.
 * first choose internal angle (= combinatorial rotation number) : 1/p
 * compute pair of parameter rays, angles of the wake : $$( \frac{1}{2^p-1} ; \frac{2}{2^p-1} )$$
 * compute list of p external angles of dynamic rays : $$( \frac{1}{2^p-1} ... \frac{n}{2^p-1} )$$

Note that :
 * internal ray 0/1 = 1/1
 * internal angle 1/p means that ray goes from period 1 component ( parent period = 1) to period p component ( child period = p)
 * as child period grows computations are harder
 * exponential growth of $$2^p$$. One can easly create a numeric value that is too large to be represented within the available storage space (  integer overflow ). For example  $$2^{34}$$ is to big for short ( 16 bit ) and long ( 32 bit) integer.

n/p
It is not so simple, as in 1/p case, to compute orbit portrait of parabolic fixed point.

Algorithm :
 * choose child period p
 * compute internal angle ( rational number) = n/p ( where n<p and n/p is ... ( to do ))
 * compute angle of the wake
 * switch to dynamic plane : use one angle from pair of parameter rays ( rays with the same angles land on the parabolic fixed point) to compute orbit portrait of parabolic fixed point

Methods for computing anglew of the wake:
 * Combinatorial algorithm = Devaney's method
 * step method:
 * compute denominator of external angle = $$2^p-1$$
 * find parameter rays that land on the root point which is on the boundary of main cardioid :
 * compute all pairs for periods 1-p
 * remove pairs which land not on the main cardioid ( inside < 1/3; 2/3 > wake )
 * compute pairs of external angles which are not inside pairs of lower periods (see image on the right )
 * choose n-th pair of angles which land on the root point

Child period 5
From all 15 period five components only 4 components are directly connected to the main cardioid

Child period 7
From all 63 period seven components only 6 components are directly connected to the main cardioid

Child period 11
Maxima CAS code using doubling map: (%i1) m(n,d):=mod(2*n,d)/d $

(%i2) m(681,2047); 1362 (%o2) 2047 (%i3) m(1362,2047); 677 (%o3) 2047 (%i4) m(677,2047); 1354 (%o4) 2047 (%i5) m(1354,2047); 661 (%o5) 2047 (%i6) m(661,2047); 1322 (%o6) 2047 (%i7) m(1322,2047); 597 (%o7) 2047 (%i8) m(597,2047); 1194 (%o8) 2047 (%i9) m(1194,2047); 341 (%o9) 2047 (%i10) m(341,2047); 682 (%o10) 2047 (%i11) m(682,2047); 1364 (%o11) 2047 (%i12) m(1364,2047); 681 (%o12) 2047 (%i13) m(681,2047); 1362 (%o13) 2047 Other version : kill(all); remvalue(all);

DoublingMap(r):= block([d,n], n:ratnumer(r), d:ratdenom(r), mod(2*n,d)/d)$

GiveOrbit(r0,OrbitLength):= block( [r,Orbit], r:r0, Orbit:[r], for i:1 thru OrbitLength step 1 do        ( r:DoublingMap(r), Orbit:endcons(r,Orbit)), return(sort(Orbit))

)$

r0: 681/2047$ period : 11; GiveOrbit(r0,period); 341  597   661   677   681   681   682   1194  1322  1354  1362  1364 (%o6) [,, , , , , , , , , , ] 2047 2047  2047  2047  2047  2047  2047  2047  2047  2047  2047  2047 float(%o6); (%o7) [0.1665852467024914, 0.2916463116756229, 0.3229115779189057, 0.3307278944797264, 0.3326819736199316, 0.3326819736199316, 0.3331704934049829, 0.5832926233512458, 0.6458231558378115, 0.6614557889594529, 0.6653639472398633, 0.6663409868099658] (%i8)

child period 21
See also wake 10/21

Maxima 5.45.1 https://maxima.sourceforge.io using Lisp GNU Common Lisp (GCL) GCL 2.6.12 Distributed under the GNU Public License. See the file COPYING. Dedicated to the memory of William Schelter. The function bug_report provides bug reporting information. (%i1) bash("r.mac"); (%o1)                            bash(r.mac) (%i2) load("r.mac"); (%o0)                               r.mac (%i1) batch("r.mac");

read and interpret /home/a/Dokumenty/maxima/rotation/r.mac (%i2) kill(all) (%o0)                               done (%i1) remvalue(all) (%o1)                                [] (%i2) DoublingMap(r):=block([d,n],n:ratnumer(r),d:ratdenom(r),mod(2*n,d)/d) (%i3) GiveOrbit(r0,OrbitLength):=block([r,Orbit],r:r0,Orbit:[r],               for i thru OrbitLength do                    (r:DoublingMap(r),Orbit:endcons(r,Orbit)),                return(sort(Orbit))) (%i4) r0:699049/2097151 (%i5) period:21 (%i6) GiveOrbit(r0,period)

349525  611669   677205   693589   697685   698709   698965   699029 (%o6) [---, ---, ---, ---, ---, ---, ---, ---, 2097151 2097151  2097151  2097151  2097151  2097151  2097151  2097151 699045   699049   699049   699050   1223338  1354410  1387178  1395370 ---, ---, ---, ---, ---, ---, ---, ---, 2097151  2097151  2097151  2097151  2097151  2097151  2097151  2097151

1397418 1397930  1398058  1398090  1398098  1398100 ---, ---, ---, ---, ---, ---] 2097151  2097151  2097151  2097151  2097151  2097151

(%o7)              /home/a/Dokumenty/maxima/rotation/r.mac

(%i9) float(%o6); (%o9) [0.166666587193769, 0.2916666467984423, 0.3229166616996106, 0.3307291654249026, 0.3326822913562257, 0.3331705728390564, 0.3332926432097641, 0.333323160802441, 0.3333307902006102, 0.3333326975501525, 0.3333326975501525, 0.3333331743875381, 0.5833332935968846, 0.6458333233992212, 0.6614583308498053, 0.6653645827124514, 0.6663411456781129, 0.6665852864195282, 0.666646321604882, 0.6666615804012205, 0.6666653951003051, 0.6666663487750763] (%i10)

result without sort(Orbit) 699049  1398098  699045   1398090  699029   1398058  698965 (%o6)/R/ [---, ---, ---, ---, ---, ---, ---, 2097151 2097151  2097151  2097151  2097151  2097151  2097151 1397930  698709   1397418  697685   1395370  693589   1387178  677205 ---, ---, ---, ---, ---, ---, ---, ---, 2097151  2097151  2097151  2097151  2097151  2097151  2097151  2097151 1354410  611669   1223338  349525   699050   1398100  699049 ---, ---, ---, ---, ---, ---, ---] 2097151  2097151  2097151  2097151  2097151  2097151  2097151 (%o7)               /home/a/Dokumenty/maxima/rotation/r.mac float(%o6); (%o8) [0.3333326975501525, 0.6666653951003051, 0.3333307902006102, 0.6666615804012205, 0.333323160802441, 0.666646321604882, 0.3332926432097641, 0.6665852864195282, 0.3331705728390564, 0.6663411456781129, 0.3326822913562257, 0.6653645827124514, 0.3307291654249026, 0.6614583308498053, 0.3229166616996106, 0.6458333233992212, 0.2916666467984423, 0.5833332935968846, 0.166666587193769, 0.3333331743875381, 0.6666663487750763, 0.3333326975501525]

Child period 27
internal angle 13/27
 * c = -0.739880396515927 +0.115700424748225 i
 * fixed point alpha z = -0.496619178870972 +0.058046457062615 i
 * period 27

The 13/27-wake of the main cardioid is bounded by the parameter rays with the angles
 * 44739241/134217727 or  p010101010101010101010101001
 * 44739242/134217727 or  p010101010101010101010101010

Child period 34
See image by Arnaud Cheritat

Angles of the wake external rays ( on parameter plane )  in different formats :

$$4985538889/17179869183 = \frac{4985538889_{10}}{17179869183_{10}} =   p0100101001001010010100100101001001 = .(0100101001001010010100100101001001) = 0.(0100101001001010010100100101001001) = 0.\overline{0100101001001010010100100101001001}_2 = \frac{100101001001010010100100101001001_2}{1111111111111111111111111111111111_2}$$

$$4985538890/17179869183 =   p0100101001001010010100100101001010 = .(0100101001001010010100100101001010) = 0.(0100101001001010010100100101001010)$$

There are 8 589 869 055 components of period 34.

External angle landing on the root points :

$$ e = \frac{n}{d} $$

where denominator d is :

$$ d= 2^{34}-1 = 17\ 179\ 869\ 183 $$

Widest sector ( which incudes critical component ) is : ( 2492769445/17179869183; 11082704036/17179869183 )

Last componet = component to the left of component including critical point zcr = 0.0. This component is almost not changing when iPeriodChild is increasing, see this video

34/89


See image by Arnaud Cheritat

Let's find some info using program Mandel by Wolf Jung : t = 34/89 = 0,382022472 // internal angle = rotational number c = -0.390837354761211 +0.586641524321638 i // Parameter c z = -0.368804231870311  +0.337614334047815 i // fixed point alfa

denominator or external angle ( computed with this program ) :

$$ d = (2^{89}-1)_{10} = 618970019642690137449562111_{10} = 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111_2$$

Location of root point using book program by Claude Heiland-Allen: 1/3 = 0.3333 3/8 = 0,375 8/21 = 0,380952381 21/55 = 0,381818182 34/89 = 0,382022472 13/34 = 0,382352941 5/13 = 0,384615385 2/5 = 0.4 1/2 = 0.5

External angles of rays that land on the root point one can compute with book program by Claude Heiland-Allen :

./mandelbrot_external_angles 53 -3.9089629378291085e-01 5.8676031775674931e-01 89 .(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001) .(01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010)

Converting to other forms using gmp :

decimal fraction = 179622968672387565806504265 / 618970019642690137449562111 decimal canonical form = 179622968672387565806504265/618970019642690137449562111 binary fraction = 01001010010010100101001001010010010100101001001010010100100101001001010010100100101001001/11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 decimal floating point number : 0.290196557138708685358212600555

decimal fraction = 179622968672387565806504266 / 618970019642690137449562111 decimal canonical form = 179622968672387565806504266/618970019642690137449562111 binary fraction = 01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010/11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 decimal floating point number : 0.290196557138708685358212602171

Note that difference in floating form of external angles :

0.290 196 557 138 708 685 358 212 602 171 0.290 196 557 138 708 685 358 212 600 555

is on 27-th decimal digit after decimal sign

0.(010 010 100 100 101 001 010 010 010 100 100 101 001 010 010 010 100 101 001 001 010 010 010 100 101 001 001 010 010 01) 0.(010 010 100 100 101 001 010 010 010 100 100 101 001 010 010 010 100 101 001 001 010 010 010 100 101 001 001 010 010 10)

and on 88-th binary digit after decimal sign.

Numerators of the orbit portrait ( external angles of rays landing on the fixed point alfa ) : 179622968672387565806504265 179622968672387565806504265 359245937344775131613008530 99521855046860125776454949 199043710093720251552909898 398087420187440503105819796 177204820732190868762077481 354409641464381737524154962 89849263286073337598747813 179698526572146675197495626 359397053144293350394991252 99824086645896563340420393 199648173291793126680840786 399296346583586253361681572 179622673524482369273801033 359245347048964738547602066 99520674455239339645642021 199041348910478679291284042 398082697820957358582568084 177195375999224579715574057 354390751998449159431148114 89811484354208181412734117 179622968708416362825468234 359245937416832725650936468 99521855190975313852310825 199043710381950627704621650 398087420763901255409243300 177204821885112373368924489 354409643770224746737848978 89849267897759356026135845 179698535795518712052271690 359397071591037424104543380 99824123539384710759524649 199648247078769421519049298 399296494157538843038098596 179622968672387548626635081 359245937344775097253270162 99521855046860057056978213 199043710093720114113956426 398087420187440228227912852 177204820732190319006263593 354409641464380638012527186 89849263286071138575492261 179698526572142277150984522 359397053144284554301969044 99824086645878971154375977 199648173291757942308751954 399296346583515884617503908 179622673524341631785445705 359245347048683263570891410 99520674454676389692220709 199041348909352779384441418 398082697818705558768882836 177195375994720980088203561 354390751989441960176407122 89811484336193782903252133 179622968672387565806504266 359245937344775131613008532 99521855046860125776454953 199043710093720251552909906 398087420187440503105819812 177204820732190868762077513 354409641464381737524155026 89849263286073337598747941 179698526572146675197495882 359397053144293350394991764 99824086645896563340421417 199648173291793126680842834 399296346583586253361685668 179622673524482369273809225 359245347048964738547618450 99520674455239339645674789 199041348910478679291349578 398082697820957358582699156 177195375999224579715836201 354390751998449159431672402 89811484354208181413782693 179622968708416362827565386 359245937416832725655130772 99521855190975313860699433 199043710381950627721398866 398087420763901255442797732 177204821885112373436033353 354409643770224746872066706 89849267897759356294571301 179698535795518712589142602 359397071591037425178285204 99824123539384712907008297 199648247078769425814016594 399296494157538851628033188

=Code=
 * program Mandel by Wolf Jung: MainMenu\Rays\Orbit Portrait, input first angle of dynamic ray ( or wake angle)

=References=