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

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);

In the elephant valley^[1][2] ( 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 :${\displaystyle (\frac{1}{2^p-1};\frac{2}{2^p-1})}$
• compute list of p external angles of dynamic rays :${\displaystyle (\frac{1}{2^p-1}...\frac{n}{2^p-1})}$

internal angle of main cardioid	parameter c = root point	parameter rays	parabolic fixed point z	dynamic rays
0/1		(0/1; 1/1)		(1/1)
1/2		(1/3; 2/3)		(1/3; 2/3)
1/3		(1/7; 2/7)		(1/7; 2/7; 4/7)
1/4		(1/15; 2/15)		(1/15; 2/15; 4/15; 8/15)
1/5		(1/31; 2/31)		(1/31; 2/31; 4/31; 8/31; 16/31)
1/6		(1/63; 2/63)		(1/63; 2/63; 4/63, 8/63, 16/33; 32/63)
1/7		(1/127; 2/127)		(1/127; 2/127; 4/127, 8/127, 16/127; 32/127; 64/127)
1/p		${\displaystyle (\frac{1}{2^p-1};\frac{2}{2^p-1})}$		${\displaystyle (\frac{1}{2^p-1}...\frac{n}{2^p-1})}$

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^[3] of${\displaystyle 2^p}$. One can easly create a numeric value that istoo large to be represented within the available storage space (integer overflow^[4] ). For example${\displaystyle 2^{34}}$ is to big for short ( 16 bit ) and long ( 32 bit) integer.

• from period 1 to period ...
• 
  1/1 =cauliflower
• 
  1/2 =San Marco fractal^[5]
• 
  1/3 = Douady fat rabbit^[6]
• 
  1/4
• 
  1/5
• 
  1/7
• 
  1/10
• 
  1/15
• 
  1/20
• 
  1/30

It is not so simple, as in 1/p case, to compute orbit portrait^[7] 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 =${\displaystyle 2^p-1}$
• 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

From all 15 period five components only 4 components are directly connected to the main cardioid^[8]

From all 63 period seven components only 6 components are directly connected to the main cardioid^[9]

• 
  parameter plane t=5/11
• 
  Dynamic plane t=5/11

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  2047float(%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)

See alsowake 10/21

Maxima 5.45.1 https://maxima.sourceforge.iousing Lisp GNU Common Lisp (GCL) GCL 2.6.12Distributed 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  20971511397418  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  20971511397930  698709   1397418  697685   1395370  693589   1387178  677205-------, -------, -------, -------, -------, -------, -------, -------, 2097151  2097151  2097151  2097151  2097151  2097151  2097151  20971511354410  611669   1223338  349525   699050   1398100  699049-------, -------, -------, -------, -------, -------, -------]2097151  2097151  2097151  2097151  2097151  2097151  2097151(%o7)               /home/a/Dokumenty/maxima/rotation/r.macfloat(%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]

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

See image by Arnaud Cheritat^[10]

Angles of the wake external rays ( on parameter plane ) in different formats :^[11]

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

${\displaystyle 4985538890/17179869183=p0100101001001010010100100101001010=.(0100101001001010010100100101001010)=0.(0100101001001010010100100101001010)}$

There are 8 589 869 055 components of period 34.

External angle landing on the root points :

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

where denominator d is :

${\displaystyle d=2^{34}-1=17179869183}$

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 , seethis video

See image by Arnaud Cheritat^[12]

Let's find some info usingprogram Mandel by Wolf Jung :

 t = 34/89 = 0,382022472 // internal angle = rotational number  c = -0.390837354761211  +0.586641524321638 i // Parameter cz = -0.368804231870311  +0.337614334047815 i // fixed point alfa

denominator or external angle (computed with this program ) :

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

Location of root point usingbook program by Claude Heiland-Allen:

1/3 = 0.33333/8 = 0,3758/21 = 0,38095238121/55 = 0,38181818234/89 = 0,38202247213/34 = 0,3823529415/13 = 0,3846153852/5 = 0.41/2 = 0.5

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

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

Converting to other forms usinggmp :

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

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

Note that difference infloating 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 ofthe orbit portrait ( external angles of rays landing on the fixed point alfa ) :

179622968672387565806504265179622968672387565806504265359245937344775131613008530995218550468601257764549491990437100937202515529098983980874201874405031058197961772048207321908687620774813544096414643817375241549628984926328607333759874781317969852657214667519749562635939705314429335039499125299824086645896563340420393199648173291793126680840786399296346583586253361681572179622673524482369273801033359245347048964738547602066995206744552393396456420211990413489104786792912840423980826978209573585825680841771953759992245797155740573543907519984491594311481148981148435420818141273411717962296870841636282546823435924593741683272565093646899521855190975313852310825199043710381950627704621650398087420763901255409243300177204821885112373368924489354409643770224746737848978898492678977593560261358451796985357955187120522716903593970715910374241045433809982412353938471075952464919964824707876942151904929839929649415753884303809859617962296867238754862663508135924593734477509725327016299521855046860057056978213199043710093720114113956426398087420187440228227912852177204820732190319006263593354409641464380638012527186898492632860711385754922611796985265721422771509845223593970531442845543019690449982408664587897115437597719964817329175794230875195439929634658351588461750390817962267352434163178544570535924534704868326357089141099520674454676389692220709199041348909352779384441418398082697818705558768882836177195375994720980088203561354390751989441960176407122898114843361937829032521331796229686723875658065042663592459373447751316130085329952185504686012577645495319904371009372025155290990639808742018744050310581981217720482073219086876207751335440964146438173752415502689849263286073337598747941179698526572146675197495882359397053144293350394991764998240866458965633404214171996481732917931266808428343992963465835862533616856681796226735244823692738092253592453470489647385476184509952067445523933964567478919904134891047867929134957839808269782095735858269915617719537599922457971583620135439075199844915943167240289811484354208181413782693179622968708416362827565386359245937416832725655130772995218551909753138606994331990437103819506277213988663980874207639012554427977321772048218851123734360333533544096437702247468720667068984926789775935629457130117969853579551871258914260235939707159103742517828520499824123539384712907008297199648247078769425814016594399296494157538851628033188

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

1. ↑muency : elephant valley
2. ↑Visual Guide To Patterns In The Mandelbrot Set by Miqel
3. ↑integer number in wikipedia
4. ↑Integer overflow in wikipedia
5. ↑planetmath : San Marco fractal
6. ↑wikipedia : Douady rabbit
7. ↑wikipedia : orbit portrait
8. ↑Parameter rays of root points of period p components by A Majewski
9. ↑Parameter rays of root points of period p components by A Majewski
10. ↑Arnaud Cheritat - gallery
11. ↑knowledgedoor calculators: convert_a_ratio_of_integers
12. ↑Arnaud Cheritat - gallery

• Book:Fractals