Fractals/Iterations in the complex plane/Mandelbrot set/chaotic part

Structure of Mandelbrot set's real slice
 * Periodic part
 * chaotic part = Main antenn

Chaotic part consist of chaotic bands deliminated by primary separators ( Misiurewicz points )

= ordering and sequences of hyperbolic componnets=

ordering
ordering of hyperbolic componnets
 * Sharkovsky ordering shows
 * ordering of hyperbolic componnets along real axis ( real slice
 * order of chaotic bands and inside periodic part
 * inside chaotic band
 * period of the biggest island inside chaotic band ( first number )
 * sequence of smaller islands periods ( next numbers)
 * other orders inside chotic bands

Sharkovsky sequence inside chaotic bands
= Separators=

primary separator

 * band-merging point = point merging 2 chaotic bands $$ B_{i-1}$$ and $$ B_i$$ are Misiurewicz points $$M_{k,p}$$

$$ m_i = M_{2^i, 2^{i-1}}$$

with
 * $$k = 2^i $$ = preperiod of critical value
 * $$p = 2^{i-1}$$ = period of critical value = period of critical point

Examples:
 * $$ c = m_0 = M_{ 1, 1} \ \ =    -2$$ = tip of main antenna, external angle = 1/2
 * $$ c = m_1 = M_{ 2, 1} \ \ =    -1.543 689 012 692 076 ...$$, external angles 5 and 7/12
 * $$ c = m_2 = M_{ 4, 2} \ \ =    -1.430 357 632 451 307 ...$$, external angles 33 and 47/80
 * $$ c = m_3 = M_{ 8, 4} \ \ =    -1.407 405 118 164 702 ...$$,  external angles 1795 and 2557/4352
 * $$ c = m_4 = M_{16, 8} \ =      -1.402 492 176 358 564...$$
 * $$ c = m_5 = M_{32, 16} =       -1.401 441 494 253 588...$$
 * $$ c = m_6 = M_{64, 32}  =      -1.401 216 504 309 415..$$
 * $$ c = m_7 = M_{128, 64} =      -1.401 168 320 839 301..$$
 * $$ c = m_8 = M_{256, 128} =     -1.401 158 001 505 211..$$
 * $$ c = m_9 = M_{512, 256} =     -1.401 155 791 424 613  ..$$
 * $$ c = m_{10} = M_{1024, 512} = -1.401 155 318 093 230  ..$$
 * $$ c = m_{11} = M_{2048, 1024} = -1.401 155 216 720 152 ..$$
 * $$ c = m_{\infty} = M_{\infty, \infty } = -1.401 155 189 0 ... ...$$ = Feigenbaum point = MF = the Myrberg-Feigenbaum point
 * $$ c = m_{\infty} = M_{\infty, \infty } = -1.401 155 189 0 ... ...$$ = Feigenbaum point = MF = the Myrberg-Feigenbaum point

So the Myrberg-Feigenbaum point is a limit of a series of band-merging points $$ m_n $$:


 * $$ \lim_{n \to \infty} m_n = m_{\infty} = c_F$$

In the text form:

Differences $$ d = m_i - m_{i+1}$$ or width of the band:
 * m[ 0]-m[ 1]| = 0.4563109873079241 = 4.5631098730792408e-01
 * m[ 1]-m[ 2]| = 0.1133313802407689 = 1.1333138024076894e-01
 * m[ 2]-m[ 3]| = 0.0229525142866049 = 2.2952514286604941e-02
 * m[ 3]-m[ 4]| = 0.0049129418061380 = 4.9129418061379937e-03
 * m[ 4]-m[ 5]| = 0.0010506821049761 = 1.0506821049760529e-03
 * m[ 5]-m[ 6]| = 0.0002249899441731 = 2.2498994417308182e-04
 * m[ 6]-m[ 7]| = 0.0000481834701138 = 4.8183470113816185e-05
 * m[ 7]-m[ 8]| = 0.0000103193340901 = 1.0319334090125309e-05
 * m[ 8]-m[ 9]| = 0.0000022100805979 = 2.2100805978642057e-06
 * m[ 9]-m[10]| = 0.0000004733313832 = 4.7333138319949342e-07
 * m[10]-m[11]| = 0.0000001013730779 = 1.0137307793200989e-07
 * m[11]- MF| = 0.0000000276268373 = 2.762684e-08

Ratio $$ r_{i-1} = \frac{m_{i-1} - m_{i-2}}{m_i - m_{i-1}}$$ tends to first Feigenbaum constant r = 4.0263428040716152e+00 r = 4.9376455592452890e+00 r = 4.6718473762357968e+00 r = 4.6759545849978759e+00 r = 4.6699069544538263e+00 r = 4.6694425212966957e+00 r = 4.6692421907265809e+00 r = 4.6692116568498836e+00 r = 4.6692035988087657e+00 r = 4.6692020490583603e+00

secondary separator

 * tree separators ( tree is a subset of band )

$$ m_{i,j} = M_{(j+1)2^i - 1, 2^i}$$

=Chaotic bands=

B0
First chotic band = B0



primary separators ( Misiurewicz points )
 * $$ c = m_0 = M_{ 1, 1} \ \ =    -2$$ = tip of main antenna, external angle = 1/2
 * $$ c = m_1 = M_{ 2, 1} \ \ =    -1.543 689 012 692 076 ...$$, external angles 5 and 7/12


 * $$m_0 \prec B_0  \prec m_1$$

0 Sharkovsky sequence

 * $$m_0 \prec (2k+1)  \prec m_1$$


 * $$m_0 \prec 3 \prec 5 \prec 7 \prec ... \prec  m_1$$

Bs
s-th chotic band = $$B_s$$


 * $$m_s \prec B_s  \prec m_{s+1}$$

s-th Sharkovsky sequence

 * $$m_s \prec (2k+1)*2^s  \prec m_{s+1}$$


 * $$m_s \prec 3*2^s \prec 5*2^s \prec 7*2^s \prec ... \prec  m_{s+1}$$

=Islands= Islands



real
real component centers of Mandelbrot Set (period less than 16) by Jay R. Hill

probably computed using :
 * FINDPERC.EXE : uses 80-bit floating point arithmetic to achieve 19 digit results.
 * FINDPERM : uses 57 digits with the extended arithmetic
 * package MFLOAT by Friedrich Kaufmann. It displays results to 37 digits.

period index root 2     1 -1.000000000000000000  3      1 -1.754877666246692760  4      1 -1.940799806529484750  4      2 -1.310702641336832880  5      1 -1.985424253054205310  5      2 -1.860782522204854870  5      3 -1.625413725123303730  6      1 -1.996376137711193750  6      2 -1.966773216392928680  6      3 -1.907280091065301960  6      4 -1.772892903381623790  6      6 -1.476014642728429890  7      1 -1.999095682327018470  7      2 -1.991814172549122210  7      3 -1.977179587006257380  7      4 -1.953705894284396240  7      5 -1.927147709363950260  7      6 -1.884803571586681790  7      7 -1.832315202751229190  7     10 -1.674066091474787970  7     11 -1.574889139752300960  8      1 -1.999774048693727320  8      2 -1.997962915597714310  8      3 -1.994332966715534900  8      4 -1.988793274309471250  8      5 -1.981655786276044430  8      6 -1.972199838366875690  8      7 -1.960758987197428950  8      8 -1.941782090318198160  8     10 -1.917098277113422000  8     11 -1.896917994678832840  8     12 -1.870003880828765360  8     13 -1.851730049410641290  8     14 -1.810001385728012070  8     19 -1.711079470013152140  8     22 -1.521817231671250990  8     23 -1.381547484432061460  9      1 -1.999943521765674000  9      2 -1.999491438016398090  9      3 -1.998586588842206870  9      4 -1.997223024696578670  9      5 -1.995419032661308130  9      6 -1.993130254878976070  9      7 -1.990376381055951220  9      8 -1.987004347515046850  9      9 -1.983810249999715080  9     10 -1.979457504855952090  9     11 -1.974780858900119900  9     12 -1.969419120730898370  9     13 -1.964024336820145270  9     14 -1.957325050535698610  9     15 -1.949574903249391220  9     18 -1.932243966576093880  9     19 -1.922285778246295770  9     20 -1.911444631473453430  9     21 -1.903116773015596850  9     22 -1.890775424360235070  9     23 -1.878382601500096120  9     26 -1.841288561509692930  9     27 -1.822756322492292640  9     28 -1.785865646410673530  9     38 -1.690142263118863300  9     39 -1.656132562574207450  9     42 -1.595680963439745770  9     43 -1.555282700768583180 10      1 -1.999985881140392100 10      2 -1.999872911766329140 10      3 -1.999646917733272930 10      4 -1.999307504280607550 10      5 -1.998855765981215220 10      6 -1.998288907350478770 10      7 -1.997608093863517390 10      8 -1.996804552742464270 10      9 -1.995924151930753200 10     10 -1.994889199350771110 10     11 -1.993747864246046470 10     12 -1.992479382178490760 10     13 -1.991120876123003380 10     14 -1.989600586912049630 10     15 -1.987940574280532540 10     16 -1.985482320973162620 10     18 -1.982719408297867730 10     19 -1.980577173424231090 10     20 -1.978293354476982960 10     21 -1.976042168611755780 10     22 -1.973497417321997840 10     23 -1.970857889101421060 10     24 -1.967742776293466570 10     25 -1.965821785502075820 10     26 -1.962378689702482390 10     27 -1.959098046552301780 10     28 -1.955423327026315770 10     29 -1.951899996873747210 10     30 -1.946873454179581380 10     35 -1.935390734772530910 10     36 -1.929320330944947580 10     37 -1.925034238501073900 10     38 -1.919635489318563350 10     39 -1.914479863678402620 10     42 -1.899832337791560340 10     43 -1.894002022860501320 10     44 -1.887172206397703060 10     45 -1.882407933753996170 10     46 -1.874314683806341270 10     47 -1.861558413753725570 10     51 -1.846627211198968380 10     52 -1.835158845261234550 10     53 -1.829509623956445570 10     54 -1.816294855238395870 10     55 -1.802436134811709390 10     74 -1.721915099588692480 10     75 -1.701700891209301220 10     78 -1.629432500446322830 10     86 -1.536243271282310500 10     87 -1.501716839409633130 10     90 -1.447008840607475070 11      1 -1.999996470335008680 11      2 -1.999968231709747580 11      3 -1.999911750208503700 11      4 -1.999826999612340710 11      5 -1.999714045769217260 11      6 -1.999572711292138060 11      7 -1.999403065044500460 11      8 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-1.892340207089897870 11     87 -1.889147135040407790 11     90 -1.880248365087264960 11     91 -1.876492993166419600 11     92 -1.871708441054241560 11     93 -1.868080148990981150 11    100 -1.853761460474161530 11    101 -1.849920890504228130 11    102 -1.843795221773942920 11    103 -1.838682168652191830 11    106 -1.825696028265308230 11    107 -1.819821501761180350 11    108 -1.811769417172880500 11    109 -1.808205447658229440 11    110 -1.796054807927162150 11    147 -1.732006272869656130 11    150 -1.695239312097040780 11    151 -1.684517123596237970 11    154 -1.663112608365032910 11    155 -1.648713821405736690 11    166 -1.604313501059346050 11    167 -1.586804684708466700 11    170 -1.563629631087201180 11    171 -1.547903761803955020 12      1 -1.999999117587260820 12      2 -1.999992058194425320 12      3 -1.999977939093294230 12      4 -1.999956758546406900 12      5 -1.999928520516907940 12      6 -1.999893213728352300 12      7 -1.999850842236313150 12      8 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-1.994468338311266000 12     41 -1.994196280539621360 12     42 -1.993897700834003610 12     43 -1.993596318489312970 12     44 -1.993285559762791820 12     45 -1.992973069476001450 12     46 -1.992645724083767440 12     47 -1.992309541878952950 12     48 -1.991929121793397660 12     49 -1.991700418739953850 12     50 -1.991304294293602250 12     51 -1.990937061962790200 12     52 -1.990561687511709180 12     53 -1.990189291729953400 12     54 -1.989797914696203210 12     55 -1.989400474495348860 12     56 -1.988979896182892350 12     57 -1.988605079892900020 12     58 -1.988161256617442870 12     59 -1.987715995936635080 12     60 -1.987243879034155150 12     61 -1.986752494228893930 12     62 -1.986136191235151480 12     67 -1.984753209551034390 12     68 -1.984088702632017470 12     69 -1.983540841872044280 12     70 -1.982988615349141780 12     71 -1.982449683309683530 12     72 -1.981880085194780060 12     73 -1.981431958985322020 12     74 -1.980853663192828480 12     75 -1.980300030250933620 12     76 -1.979729378401139660 12     77 -1.979183607438254600 12     78 -1.978591171694646240 12     79 -1.977985134233276210 12     82 -1.976364477321859740 12     83 -1.975726245926107480 12     84 -1.975081487961333710 12     85 -1.974479154356912780 12     86 -1.973821883630731810 12     87 -1.973168645674608590 12     88 -1.972443394067073310 12     89 -1.971956620639514910 12     90 -1.971207317826686790 12     91 -1.970506316020478710 12     92 -1.969766673874645700 12     93 -1.969063069240782970 12     94 -1.968223161521319060 12     95 -1.966808803425221040 12     99 -1.965318943530410270 12    100 -1.964412302186846800 12    101 -1.963641270444242080 12    102 -1.962790403216217370 12    103 -1.961962210760019850 12    104 -1.961002493823277540 12    105 -1.960517451508275110 12    106 -1.959533985762773850 12    107 -1.958661628575311190 12    108 -1.957738162895568380 12    109 -1.956906866402774140 12    110 -1.955921061804021570 12    111 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=See also=
 * real slice of Mandelbrot set
 * JAKOBSON’S THEOREM
 * proof by Jean Christophe Yoccoz
 * math.stackexchange question: m-set-interior-point-probability-on-the-real-axis

=References=