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 )^[1]

ordering and sequences of hyperbolic componnets

Chaotic bands from B0 to B10

ordering

ordering of hyperbolic componnets^[2]

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

sequences

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: ^[3]

$c=m_{0}=M_{1,1}\ \ =-2$ = tip of main antenna , external angle = 1/2^[4]
$c=m_{1}=M_{2,1}\ \ =-1.543689012692076...$ , external angles 5 and 7/12
$c=m_{2}=M_{4,2}\ \ =-1.430357632451307...$ , external angles 33 and 47/80
$c=m_{3}=M_{8,4}\ \ =-1.407405118164702...$ , external angles 1795 and 2557/4352
$c=m_{4}=M_{16,8}\ =-1.402492176358564...$
$c=m_{5}=M_{32,16}=-1.401441494253588...$
$c=m_{6}=M_{64,32}=-1.401216504309415..$
$c=m_{7}=M_{128,64}=-1.401168320839301..$
$c=m_{8}=M_{256,128}=-1.401158001505211..$
$c=m_{9}=M_{512,256}=-1.401155791424613..$
$c=m_{10}=M_{1024,512}=-1.401155318093230..$
$c=m_{11}=M_{2048,1024}=-1.401155216720152..$
...
$c=m_{\infty }=M_{\infty ,\infty }=-1.4011551890......$ = 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:

double m[12] = {
-2.0,
-1.543689012692076,
-1.430357632451307,
-1.407405118164702,
-1.402492176358564,
-1.401441494253588,
-1.401216504309415,
-1.401168320839301,
-1.401158001505211,
-1.401155791424613,
-1.401155318093230,
-1.401155216720152  

};

const complex double cf = -1.401155189093314712; //the Feigenbaum point -1.401155 = m[infinity]

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

Mandelbrot set - real slice chotic band 0

primary separators ( Misiurewicz points )

$c=m_{0}=M_{1,1}\ \ =-2$ = tip of main antenna , external angle = 1/2^[5]
$c=m_{1}=M_{2,1}\ \ =-1.543689012692076...$ , 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

Wakes_along_the_main_antenna_in_the_Mandelbrot_set

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 -1.999204566491225700
 11      9 -1.998980020902538600
 11     10 -1.998724654904751430
 11     11 -1.998441414477602890
 11     12 -1.998129024076289590
 11     13 -1.997789739591220190
 11     14 -1.997419190865984310
 11     15 -1.997018796675792250
 11     16 -1.996572112247164520
 11     17 -1.996178289863562060
 11     18 -1.995673667316987960
 11     19 -1.995158050516975210
 11     20 -1.994612320312243470
 11     21 -1.994046774294736830
 11     22 -1.993442193455161600
 11     23 -1.992810279386047150
 11     24 -1.992132473419578720
 11     25 -1.991491196324180660
 11     26 -1.990750707290041020
 11     27 -1.989994261107521120
 11     28 -1.989195014772458840
 11     29 -1.988381342856646010
 11     30 -1.987484171114959340
 11     31 -1.986467602520304380
 11     34 -1.984400062463938130
 11     35 -1.983262197775561640
 11     36 -1.982173393685911000
 11     37 -1.981135356645495560
 11     38 -1.980017288110872010
 11     39 -1.978887819382171960
 11     40 -1.977643304770228910
 11     41 -1.976716792680409560
 11     42 -1.975406229576063050
 11     43 -1.974149564294258260
 11     44 -1.972824987746817520
 11     45 -1.971566130691004370
 11     46 -1.970142650136013340
 11     47 -1.968653636464047760
 11     50 -1.964858951761163740
 11     51 -1.963211626815652330
 11     52 -1.961521263299723720
 11     53 -1.959989860019309700
 11     54 -1.958208072391217650
 11     55 -1.956414775411862780
 11     56 -1.954165857400134740
 11     57 -1.953255117397111030
 11     58 -1.950735248729150630
 11     59 -1.948330193383247060
 11     60 -1.944985211053557280
 11     69 -1.937494133795147620
 11     70 -1.933711217574897610
 11     71 -1.930830967476639730
 11     74 -1.923587538548829290
 11     75 -1.921000946645860180
 11     76 -1.918185148739355740
 11     77 -1.916003483251741820
 11     78 -1.912969655532352200
 11     79 -1.909605827581144710
 11     82 -1.905032905698242700
 11     83 -1.901492571473506980
 11     84 -1.898089027919550120
 11     85 -1.895754220010477820
 11     86 -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 -1.999801372159398660
 12      9 -1.999744977573602180
 12     10 -1.999681361196536870
 12     11 -1.999610705772966920
 12     12 -1.999532931591664470
 12     13 -1.999448178159224800
 12     14 -1.999356173857820970
 12     15 -1.999257007353280670
 12     16 -1.999149746318768050
 12     17 -1.999040172549457130
 12     18 -1.998918689425355610
 12     19 -1.998791118960928310
 12     20 -1.998656364602012690
 12     21 -1.998515059138383320
 12     22 -1.998366020382166820
 12     23 -1.998209966745463170
 12     24 -1.998045797619839560
 12     25 -1.997878407120788970
 12     26 -1.997699713581369300
 12     27 -1.997514661455194640
 12     28 -1.997321668110368990
 12     29 -1.997122425527586520
 12     30 -1.996912951848569980
 12     31 -1.996692284798253200
 12     32 -1.996379698285911190
 12     34 -1.996049041875495070
 12     35 -1.995799524491821550
 12     36 -1.995546208217139120
 12     37 -1.995290333633020690
 12     38 -1.995024299424194510
 12     39 -1.994752157724171150
 12     40 -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 -1.954885643528039000
 12    114 -1.952493902282131290
 12    115 -1.951328308699483840
 12    116 -1.950113621874738610
 12    117 -1.949020027566269450
 12    118 -1.947616978282750130
 12    119 -1.946040258941837180
 12    120 -1.942519046352060340
 12    138 -1.936323613582675850
 12    139 -1.934544936920119570
 12    140 -1.932880761211511300
 12    141 -1.931619426517274190
 12    142 -1.930068671898056470
 12    143 -1.928487858679416940
 12    146 -1.925845936126932800
 12    147 -1.924314273608037780
 12    148 -1.922842861534029360
 12    149 -1.921733096087524110
 12    150 -1.920300759000663660
 12    151 -1.918955506209422470
 12    154 -1.915201572806543990
 12    155 -1.913757361466214350
 12    156 -1.912103324745894690
 12    157 -1.910742135737493880
 12    158 -1.907397618358110370
 12    164 -1.903839831363751100
 12    165 -1.902437049527024580
 12    166 -1.900627919859425650
 12    167 -1.899025573524404280
 12    170 -1.894811910350530710
 12    171 -1.893205684645489570
 12    172 -1.891398749834893900
 12    173 -1.890145185112852020
 12    174 -1.888167413897888940
 12    175 -1.885836225640821480
 12    178 -1.883801191013467910
 12    179 -1.881338223859112010
 12    180 -1.879089532912819080
 12    181 -1.877675388360729820
 12    182 -1.875374111087519000
 12    183 -1.873180865043625740
 12    186 -1.865723323816962500
 12    199 -1.856438079483190820
 12    202 -1.848042511645563640
 12    203 -1.845283918954318180
 12    204 -1.842107555871598930
 12    205 -1.840467028507122740
 12    206 -1.836970543993694430
 12    211 -1.827586466122061330
 12    212 -1.823606190692988170
 12    213 -1.821915822765622860
 12    214 -1.817969082849710430
 12    215 -1.814483842175301100
 12    218 -1.804949143729163450
 12    219 -1.799753675058546030
 12    220 -1.789175045986018750
 12    221 -1.778266821111081960
 12    294 -1.725978693774019480
 12    295 -1.717972319411030950
 12    298 -1.704840650473635820
 12    299 -1.698845700782579910
 12    302 -1.678767891298070120
 12    307 -1.669642176971863640
 12    310 -1.641278502145420340
 12    331 -1.612971248577386260
 12    342 -1.541094106491382820
 12    343 -1.530056455961480150
 12    346 -1.512452670794389080
 12    347 -1.480925450162162270
 12    359 -1.416977731250212230
 13      1 -1.999999779397058820
 13      2 -1.999998014567262920
 13      3 -1.999994484884841200
 13      4 -1.999989190234989400
 13      5 -1.999982130856013370
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15 1049  -1.7300197852838890
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15 1051  -1.7234098634226422
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15 1056  -1.7068814158545038
15 1057  -1.7027200389410262
15 1058  -1.7007226944384994
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15 1060  -1.6962870993381349
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15 1064  -1.6858580190765295
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15 1066  -1.6816142419440332
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15 1068  -1.6664847856572718
15 1069  -1.6615091413654233
15 1070  -1.6595205625849374
15 1071  -1.6526893214701635
15 1072  -1.6504972057019640
15 1073  -1.6469364429838058
15 1074  -1.6448290052337982
15 1075  -1.6324071504889990
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15 1077  -1.6063191377934081
15 1078  -1.6022550082846885
15 1079  -1.5992072842960157
15 1080  -1.5922173062740226
15 1081  -1.5890049741389880
15 1082  -1.5845353317028372
15 1083  -1.5809434433997984
15 1084  -1.5694774688662991
15 1085  -1.5658309180921859
15 1086  -1.5615070900363253
15 1087  -1.5580236785479356
15 1088  -1.5527157930206532
15 1089  -1.5495791959281349
15 1090  -1.5463628057066650
15 1091  -1.5442285601195278

References

[1] ractalforums.org : primary-separators-misiurewicz-points-of-the-mandelbrot-sets-real-slice

[2] [ Pastor96 ] = G. Pastor, M. Romera and F. Montoya, "An approach to the ordering of one-dimensional quadratic maps", Chaos, Solitons and Fractals, 7/4 (1996), 565-584. Preprint

[3] A scaling constant equal to unity in 1D- quadratic maps by M. ROMERA, G. PASTOR and F. MONTOYA

[4] G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 52-71

[5] G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 52-71

[1]

[2]

[3]

[4]

[5]

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

Contents