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

Structure of Mandelbrot set's real slice

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

# Separators

## primary separator

• band-merging point = point merging 2 chaotic bands ${\displaystyle B_{i-1}}$  and ${\displaystyle B_{i}}$  are Misiurewicz points ${\displaystyle M_{k,p}}$
 ${\displaystyle m_{i}=M_{2^{i},2^{i-1}}}$


with

• ${\displaystyle k=2^{i}}$  = preperiod of critical value
• ${\displaystyle p=2^{i-1}}$  = period of critical value = period of critical point

Examples: [3]

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

So the Myrberg-Feigenbaum point is a limit of a series of band-merging points ${\displaystyle m_{n}}$ :

${\displaystyle \lim _{n\to \infty }m_{n}=m_{\infty }=c_{F}}$

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 ${\displaystyle d=m_{i}-m_{i+1}}$  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 ${\displaystyle r_{i-1}={\frac {m_{i-1}-m_{i-2}}{m_{i}-m_{i-1}}}}$  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 )
 ${\displaystyle m_{i,j}=M_{(j+1)2^{i}-1,2^{i}}}$


# Chaotic bands

## B0

First chotic band = B0

• ${\displaystyle c=m_{0}=M_{1,1}\ \ =-2}$  = tip of main antenna , external angle = 1/2[5]
• ${\displaystyle c=m_{1}=M_{2,1}\ \ =-1.543689012692076...}$ , external angles 5 and 7/12

${\displaystyle m_{0}\prec B_{0}\prec m_{1}}$

### 0 Sharkovsky sequence

${\displaystyle m_{0}\prec (2k+1)\prec m_{1}}$
${\displaystyle m_{0}\prec 3\prec 5\prec 7\prec ...\prec m_{1}}$

## Bs

s-th chotic band = ${\displaystyle B_{s}}$

${\displaystyle m_{s}\prec B_{s}\prec m_{s+1}}$

### s-th Sharkovsky sequence

${\displaystyle m_{s}\prec (2k+1)*2^{s}\prec m_{s+1}}$
${\displaystyle m_{s}\prec 3*2^{s}\prec 5*2^{s}\prec 7*2^{s}\prec ...\prec m_{s+1}}$

# Islands

## real

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
13      6 -1.999973306030850830
13      7 -1.999962715997427380
13      8 -1.999950358622764070
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