Fractals/Iterations in the complex plane/construct

How to construct a map with desired properities ? Shape Shape of Julia set critical orbit external rays landind on the repelling cycle ( spirals) Relation between shape types and dynamics: n-th arm spiral: attracting or repelling n-periodic orbit ( cycle) closed curve: Siegel disc ( rotation) n-th arm star = period n parabolic root Modelling or shaping Usually one should controll 2 parameters: fixed point period p orbit See Shaping spirals Conversions and evolutions Examples Rational functions with prescribed critical points by I. Scherbak constructing map with Fatou components of desired typefractalforums: julia-sets-true-shape-and-escape-time Constructing polynomials whose Julia set resemble a desired shapefractalforums : constructing-polynomials-whose-julia-set-resemble-a-desired-shape Constructing polynomials that have an attracting cycle visiting predefined points in the plane fractalforums : constructing-polynomials-with-attracting-cycles constructuing QUARTIC JULIA SETS INCLUDING ANY TWO COPIES OF QUADRATIC JULIA SETS QUARTIC JULIA SETS INCLUDING ANY TWO COPIES OF QUADRATIC JULIA SETS by Koh Katagata codepen SE constructing critically finite real polynomial maps with specified combinatoricsThe W. Thurston algorithm applied to real polynomial maps Araceli Bonifant, J. Milnor, S. Sutherland Published 15 May 2020 roots The fundamental theorem of algebra states that every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex rootsThe fundamental theorem of algebra in wikipediaEd Pegg Jr "The Fundamental Theorem of Algebra" http://demonstrations.wolfram.com/TheFundamentalTheoremOfAlgebra/ Wolfram Demonstrations Project Published: November 10 2011 The factor theoremFactor theorem in wikipedia states that a polynomial has a factor if and only if (i.e. is a root of multiplicityMultiplicity_of_a_root_of_a_polynomial in wikipedia m) Examples The polynomialfractalforums.org : julia-sets-true-shape-and-escape-time thumb|right|Graph of the polynomial y = x^3 + 2*x^2 - 7*x + 4 with its roots (zeros) -4 and 1 marked. The polynomial has roots: 1 of multiplicity 2 -4 of multiplicity 1 rboyce1000 p(z) = z^4 + O(z^2), where the four roots of p(z) are: one fixed at the origin, the remaining three forming the vertices of an equilateral triangle centered at the origin and rotating. create polynomial with desired properities f(z) = z*g(z) with root at origin g(z) is a 3-rd root of unity = f(z) = z(z^2+z+1) One can check it with Maxima CAS (%i1) solve([z*(z^2+z+1)=0],[z]); sqrt(3) %i + 1 sqrt(3) %i - 1 (%o1) [z = - --------------, z = --------------, z = 0] 2 2 (%i2) to rotate it around origin let's change 1 with : wher t is a proper fraction in turns See : Vector_field short videos by boyce1000 one parameter System of 2 equations: where: is a rational function with one parameter c is the -fold composition of with itself is a cyclic point ( point of limit cycle) p is a period of the cycle is a multiplierStability_of_periodic_points_(orbit)_-_multiplier in wikipedia ( complex number) is a stability of the cycle ( Real number ) Input : function p ( integer) r ( real number) ( real number or rational number) Unknowns ( solutions or output): parameter c ( complex number) periodic point ( complex number) Maxima CAS program: /* batch("m.mac"); */ display2d:false$ kill(all)$ ratprint:false$ /* complex quadratic polynomial */ f(z,c):= z*z+c $ /* iterated function */ F(z, c, n) := if n=1 then f(z,c) else f(F(z, c, n-1),c)$ /* multiplier = first deric=vative */ m(z,c,p):= diff(F(z,c,p),z,1)$ l(r,t) := float(rectform(r*exp(2*%pi*t*%i)))$ /* input */ p:5$ r:1.0$ t:0$ /* system of equations */ e1: F(z,c,p)=z; e2: m(z,c,p)=l(r,t); /* output = solutions = 2 complex number: c, z */ s:solve([e1,e2])$ s:map('float,s)$ s:map('rectform,s); Example output: For : p = 3 r=1.0 t=0.0 [ [z = 0.5,c = 0.25], [z = (-0.4330127018922193*%i)-0.25,c = (-0.6495190528383289*%i)-0.125], [z = 0.4330127018922193*%i-0.25,c = 0.6495190528383289*%i-0.125], [z = -0.05495813133539004,c = -1.75], [z = 1.301937809824245,c = -1.75], [z = -1.746979634104245,c = -1.75] ] For : p = 5 r=1.0 t=0.0 [ [z = 0.5,c = 0.25], [z = 0.4755282581475767*%i+0.1545084971874737,c = 0.3285819450744551*%i+0.3567627457812099], [z = 0.1545084971874737-0.4755282581475767*%i, c = 0.3567627457812106-0.3285819450744586*%i], [z = 0.2938926261462365*%i-0.4045084971874737, c = 0.5316567552200239*%i-0.4817627457812153], [z = (-0.2938926261462365*%i)-0.4045084971874737, c = (-0.5316567552199369*%i)-0.481762745781224], [z = -0.003102011282477321,c = -1.985409652076318], [z = 0.0109289978340113,c = -1.860587002096436], [z = 8.008393221517376E-4*%i-0.01213161194929343, c = 1.100298437397382*%i-0.1978729466687337], [z = (-8.008393221517376E-4*%i)-0.01213161194929343, c = (-1.100298437397305*%i)-0.1978729466687667], [z = 0.02151217276434695*%i-0.005267866463337371, c = 0.3797412022535638*%i-1.256801993945385], [z = (-0.02151217276434695*%i)-0.005267866463337371, c = (-0.3797412022517599*%i)-1.256801993944077], [z = 0.02591758988716001*%i+0.0096648625988135, c = 0.9868115621249533*%i-0.04506136597934137], [z = 0.0096648625988135-0.02591758988716001*%i, c = (-0.9868115621250132*%i)-0.04506136597930513], [z = -0.02506558296814108,c = -1.624396967608546], [z = 0.02532354987824971*%i-0.0286751769590709, c = 0.6415066667139064*%i+0.3599331333357185], [z = (-0.02532354987824971*%i)-0.0286751769590709, c = 0.3599331333357186-0.6415066667139071*%i], [z = 0.7018214526647177,c = -1.860587002096436], [z = 0.5745382937725365*%i+0.1798116252110209, c = (-0.379741202251533*%i)-1.25680199394442], [z = 0.1798116252110209-0.5745382937725365*%i, c = 0.3797412022514344*%i-1.256801993944486], [z = -0.5997918293000261,c = -1.624396967608546], [z = 0.6400543521659254*%i+0.3601141169309163, c = 0.6415066667138928*%i+0.3599331333356947], [z = 0.3601141169309163-0.6400543521659254*%i, c = 0.3599331333356951-0.6415066667138929*%i], [z = 0.747361547631752*%i+0.4122389750905872, c = 0.3599331333377524-0.6415066667118048*%i], [z = 0.4122389750905872-0.747361547631752*%i,c = 0.6415066667118131*%i+0.3599331333377574], [z = -1.264646754738656,c = -1.624396967608546], [z = 0.838427461519175*%i+0.1867295812979602,c = (-0.9868115621248*%i)-0.04506136597962632], [z = 0.1867295812979602-0.838427461519175*%i, c = 0.9868115621248269*%i-0.04506136597961512], [z = 1.012227741688957,c = -1.624396967608546], [z = 0.6736931444481549*%i-0.7131540376767388, c = 0.9868115621009495*%i-0.04506136566593825], [z = (-0.6736931444481549*%i)-0.7131540376767388, c = (-0.9868115621015654*%i)-0.04506136566602404], [z = 0.6816651712455555*%i+0.8064792250322852, c = (-1.100298438532418*%i)-0.1978729463920518], [z = 0.8064792250322852-0.6816651712455555*%i,c = 1.100298438531886*%i-0.197872946387467], [z = 0.9873125420152975*%i-0.04563967787575593, c = 0.9868115621249436*%i-0.04506136597927069], [z = (-0.9873125420152975*%i)-0.04563967787575593, c = (-0.9868115621249249*%i)-0.04506136597929692], [z = -1.368033648790746,c = -1.860587002096436], [z = -1.623768668573244,c = -1.624396967608546], [z = 1.600752508361204,c = -1.860587002096436], [z = 0.8177857184842046*%i-0.8491638964763748, c = 0.6415066726649287*%i+0.3599331357137042], [z = (-0.8177857184842046*%i)-0.8491638964763748, c = 0.3599331357115682-0.6415066726792946*%i], [z = -1.860467532467532,c = -1.860586580956207], [z = 0.1585230889211015*%i+1.129895436404861, c = (-0.3797412017812437*%i)-1.256801993890818], [z = 1.129895436404861-0.1585230889211015*%i, c = 0.3797412020742688*%i-1.256801993924219], [z = 1.102491882350288*%i+0.07994573682221373, c = 0.641506666713125*%i+0.3599331333375105], [z = 0.07994573682221373-1.102491882350288*%i, c = 0.3599331333375118-0.641506666713142*%i], [z = 1.10027900645412*%i-0.1977264120044163,c = 1.100298437399976*%i-0.1978729466589521], [z = (-1.10027900645412*%i)-0.1977264120044163, c = (-1.100298437392994*%i)-0.1978729466579122], [z = 0.3795145554958574*%i-1.257237017109811, c = 0.3797412012322979*%i-1.256801993538778], [z = (-0.3795145554958574*%i)-1.257237017109811, c = (-0.3797412011893692*%i)-1.256801993401957], [z = 0.8966903093631682*%i-1.01776444141452, c = 0.986811439368143*%i-0.04506141337632084], [z = (-0.8966903093631682*%i)-1.01776444141452, c = (-0.9868114393633113*%i)-0.04506141338736716], [z = 1.407944514501891,c = -1.985409652076318], [z = 0.7215120925377011*%i+1.234881318742427, c = (-1.100298500720014*%i)-0.1978727350763138], [z = 1.234881318742427-0.7215120925377011*%i,c = 1.100298500782114*%i-0.1978727352231734], [z = 0.6651899971189704*%i-1.369391104706556, c = 1.100298438532065*%i-0.1978727774731155], [z = (-0.6651899971189704*%i)-1.369391104706556, c = (-1.100298478086625*%i)-0.1978727911942495], [z = 0.1731238730127708*%i-1.554564024233688, c = 0.3797412149717089*%i-1.256801976456581], [z = (-0.1731238730127708*%i)-1.554564024233688, c = (-0.3797411926534995*%i)-1.256801968631482], [z = 1.842105908761944,c = -1.985410334346504], [z = 1.956403762662807,c = -1.985409652076318] ] Mandelbrot Set - Convergent Evolution of P/Q Limbs in Seahorse Valley Mandelbrot Set - Convergent Evolution of P/Q Limbs in Seahorse Valley by izaytsev0 main cardioid seahorse valley = Gap between the head ( period 2 component) and the body (or shoulder = main cardioid). Particularly the upper one part 2 windows left: limbs from period 2 component right: limbs from period 1 componnet in each window one can see limb p/q from 14/30 on the right and increasing p ? Compare with Real dense fractal Zoom! Part 2 by SeryZone Arts See also perturbation math.stackexchange question: how-can-i-reconstruct-a-julia-set-by-a-given-image ? References Bishop, Christopher J. Constructing entire functions by quasiconformal folding. Acta Math. 214 (2015), no. 1, 1--60. doi:10.1007/s11511-015-0122-0. Constructing entire functions with non-locally connected Julia set by quasiconformal surgery by Yanhua Zhang, Gaofei Zhang Kumar, Rajen & Nayak, Tarakanta. (2020). The real non-attractive fixed point conjecture and beyond. Specifying attracting cycles for Newton maps of polynomials by James T. Campbell, Jared T. Collins Exploring the Mandelbrot set. The Orsay Notes. Adrien Douady John H. Hubbard: constructing a polynomial with a given tree Godillon, S'ebastien. “Construction of rational maps with prescribed dynamics.” (2010). On building polynomials, November 2005The Mathematical Gazette 89(516), DOI: 10.2307/3621936, Christopher Sangwin