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 )fractalforums.org : primary-separators-misiurewicz-points-of-the-mandelbrot-sets-real-slice ordering and sequences of hyperbolic componnets 1 800 border:none; center Chaotic bands from B0 to B10 Mandelbrot set - real slice chotic band 0.png Mandelbrot set - real slice chotic band 1.png Mandelbrot_set_-_real_slice_chotic_band_2.png Mandelbrot_set_-_real_slice_chotic_band_3.png Mandelbrot_set_-_real_slice_chotic_band_4.png Mandelbrot_set_-_real_slice_chotic_band_5.png Mandelbrot_set_-_real_slice_chotic_band_6.png Mandelbrot_set_-_real_slice_chotic_band_7.png Mandelbrot_set_-_real_slice_chotic_band_8.png Mandelbrot_set_-_real_slice_chotic_band_9.png Mandelbrot_set_-_real_slice_chotic_band_10.png ordering ordering of hyperbolic componnets[ 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 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 and are Misiurewicz points with = preperiod of critical value = period of critical value = period of critical point Examples: A scaling constant equal to unity in 1D- quadratic maps by M. ROMERA, G. PASTOR and F. MONTOYA = tip of main antenna , external angle = 1/2G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 52-71 , external angles 5 and 7/12 , external angles 33 and 47/80 , external angles 1795 and 2557/4352 ... = Feigenbaum point = MF = the Myrberg-Feigenbaum point So the Myrberg-Feigenbaum point is a limit of a series of band-merging points : 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 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 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 ) Chaotic bands B0 First chotic band = B0 thumb|right|Mandelbrot set - real slice chotic band 0 primary separators ( Misiurewicz points ) = tip of main antenna , external angle = 1/2G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 52-71 , external angles 5 and 7/12 0 Sharkovsky sequence Bs s-th chotic band = s-th Sharkovsky sequence Islands Islands center|2000px|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 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