File:Paritition of dynamic plane of quadratic polynomial for 129 over 16256.svg
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DescriptionParitition of dynamic plane of quadratic polynomial for 129 over 16256.svg |
English: Partition of dynamic plane of complex quadratic polynomial for t = 129 /16256. Julia set is a dendrite ( = connected with no interior). Parameter is a Misiurewicz point. External dynamic ray for angle t lands on the critical value z= c ( image of the critical point z = 0 ). Two external rays for the angles t/2 and (t+1)/2 land on the critical point. These rays and their landing point ( critical point z = 0) divide the dynamic plane into two components. This partition used in the definition of the kneading sequence[1] of the external angle t[2] |
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Source | Own work |
Author | Adam Majewski |
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Maxima CAS source code
/* batch file for Maxima CAS maxima batch("k.mac"); ------------ program Mandel by Wolf Jung ------------------------- The angle 3/30 or 0p0011 has preperiod = 1 and period = 4. The corresponding parameter ray is landing at a Misiurewicz point of preperiod 1 and period dividing 4. Do you want to draw the ray and to shift c to the landing point? ------------------- */ kill(all); remvalue(all); ratprint:false; /* It doesn't change the computing, just the warnings. */ display2d:false; /* --------------------------definitions of functions ------------------------------*/ f(z,c):=z*z+c; /* Complex quadratic map */ finverseplus(z,c):=float(rectform(sqrt(z-c)))$ finverseminus(z,c):=float(rectform(-sqrt(z-c)))$ /* */ fn(p, z, c) := if p=0 then z elseif p=1 then f(z,c) else f(fn(p-1, z, c),c)$ /*Standard polynomial F_p \, which roots are periodic z-points of period p and its divisors */ F(p, z, c) := fn(p, z, c) - z $ /* Function for computing reduced polynomial G_p\, which roots are periodic z-points of period p without its divisors*/ G[p,z,c]:= block( [f:divisors(p), t:1], /* t is temporary variable = product of Gn for (divisors of p) other than p */ f:delete(p,f), /* delete p from list of divisors */ if p=1 then return(F(p,z,c)), for i in f do t:t*G[i,z,c], g: F(p,z,c)/t, return(ratsimp(g)) )$ GiveRoots(g):= block( [cc], cc:bfallroots(expand(%i*g)=0), cc:map(rhs,cc),/* remove string "c=" */ cc:map('float,cc), return(cc) )$ /* Gives points of backward orbit of z=repellor problem for 1/10 */ GiveBackwardOrbit(c,repellor,zxMin,zxMax,zyMin,zyMax,iXmax,iYmax, hit_limit):= block( hit_limit:10, /* proportional to number of details and time of drawing */ PixelWidth:(zxMax-zxMin)/iXmax, PixelHeight:(zyMax-zyMin)/iYmax, /* 2D array of hits pixels . Hit > 0 means that point was in orbit */ array(Hits,fixnum,iXmax,iYmax), /* no hits for beginning */ /* choose repeller z=repellor as a starting point */ stack:[repellor], /*save repellor in stack */ /* save first point to list of pixels */ x_y:[repellor], /* reversed iteration of repellor */ loop, /* pop = take one point from the stack */ z:last(stack), stack:delete(z,stack), /*inverse iteration - first preimage (root) */ z:finverseplus(z,c), /* translate from world to screen coordinate */ iX:fix((realpart(z)-zxMin)/PixelWidth), iY:fix((imagpart(z)-zyMin)/PixelHeight), hit:Hits[iX,iY], if hit<hit_limit then ( Hits[iX,iY]:hit+1, stack:endcons(z,stack), /* push = add z at the end of list stack */ if hit=0 then x_y:endcons( z,x_y) ), /*inverse iteration - second preimage (root) */ z:-z, /* translate from world to screen coordinate, coversion to integer */ iX:fix((realpart(z)-zxMin)/PixelWidth), iY:fix((imagpart(z)-zyMin)/PixelHeight), hit:Hits[iX,iY], if hit<hit_limit then ( Hits[iX,iY]:hit+1, stack:endcons(z,stack), /* push = add z at the end of list stack to continue iteration */ if hit=0 then x_y:endcons( z,x_y) ), if is(not emptyp(stack)) then go(loop), return(x_y) /* list of pixels in the form [z1,z2] */ )$ /*-----------------------------------*/ Psi_n(r,t,z_last, Max_R):= /* */ block( [iMax:200, iMax2:0], /* ----- forward iteration of 2 points : z_last and w --------------*/ array(forward,iMax-1), /* forward orbit of z_last for comparison */ forward[0]:z_last, i:0, while cabs(forward[i])<Max_R and i< ( iMax-2) do ( /* forward iteration of z in fc plane & save it to forward array */ forward[i+1]:forward[i]*forward[i] + c, /* z*z+c */ /* forward iteration of w in f0 plane : w(n+1):=wn^2 */ r:r*2, /* square radius = R^2=2^(2*r) because R=2^r */ t:mod(2*t,1), /* */ iMax2:iMax2+1, i:i+1 ), /* compute last w point ; it is equal to z-point */ R:2^r, /* w:R*exp(2*%pi*%i*t), z:w, */ array(backward,iMax-1), backward[iMax2]:float(rectform(ev(R*exp(2*%pi*%i*t)))), /* use last w as a starting point for backward iteration to new z */ /* ----- backward iteration point z=w in fc plane --------------*/ for i:iMax2 step -1 thru 1 do ( temp:float(rectform(sqrt(backward[i]-c))), /* sqrt(z-c) */ scalar_product:realpart(temp)*realpart(forward[i-1])+imagpart(temp)*imagpart(forward[i-1]), if (0>scalar_product) then temp:-temp, /* choose preimage */ backward[i-1]:temp ), return(backward[0]) )$ /* problems for c= -2 and t = 1/2 */ GiveRay(t,c):= block( [r], /* range for drawing R=2^r ; as r tends to 0 R tends to 1 */ rMin:1E-10, /* 1E-4; rMin > 0 ; if rMin=0 then program has infinity loop !!!!! */ rMax:3, dz : 100, /* dz : cabs ( z - last_z) ; if dz is to small then loop is not ending */ MachineEpsilonDouble: 1E-16, caution:0.9330329915368074, /* r:r*caution ; it gives smaller r */ /* upper limit for iteration */ R_max:300, /* */ zz:[], /* array for z points of ray in fc plane */ /* some w-points of external ray in f0 plane */ r:rMax, while 2^r<R_max do r:2*r, /* find point w on ray near infinity (R>=R_max) in f0 plane */ R:2^r, w:float(rectform(ev(R*exp(2*%pi*%i*t)))), z:w, /* near infinity z=w */ zz:cons(z,zz), unless (r<rMin or dz < MachineEpsilonDouble) do ( /* new smaller R */ r:r*caution, R:2^r, /* */ w:float(rectform(ev(R*exp(2*%pi*%i*t)))), /* */ last_z:z, z:Psi_n(r,t,last_z,R_max), /* z=Psi_n(w) */ dz : cabs ( z - last_z), zz:cons(z,zz) ), return(zz) )$ /* converts complex number z = x*y*%i to the list in a draw format: [x,y] */ d(z):=[float(realpart(z)), float(imagpart(z))]$ ToPoints(myList):= points(map(d,myList))$ /* give Draw List from one point*/ ToPoint(z):=points([d(z)])$ /* compile(all)$ */ /* ----------------------- main ----------------------------------------------------*/ start:elapsed_run_time (); HitLimit:15$ /* proportional to number of details and time of drawing */ /* external angle in turns */ /* resolution is proportional to number of details and time of drawing */ iX_max:2000$ iY_max:2000$ /* define z-plane ( dynamical ) */ ZxMin:-2.0$ ZxMax:2.0$ ZyMin:-2.0$ ZyMax:2.0$ t:129/16256$ /* give c a value */ c: 0.397391822296541 +0.133511204871878*%i $ /* one can compute it from t */ /* compute fixed points */ Beta:float(rectform((1+sqrt(1-4*c))/2)); /* compute repelling fixed point beta */ alfa:float(rectform((1-sqrt(1-4*c))/2)); /* other fixed point */ /* compute backward orbit of repelling fixed point*/ xy: GiveBackwardOrbit(c,Beta,ZxMin,ZxMax,ZyMin,ZyMax,iX_max,iY_max, HitLimit)$ /* compute ray points & save to zz list */ eRay : GiveRay(t,c)$ eRayT:GiveRay(t/2,c)$ eRayTp:GiveRay((t+1)/2,c)$ /* time of computations */ time:fix(elapsed_run_time ()-start); /* draw it using draw package by */ load(draw)$ path:"~/maxima/batch/julia/knead/k_129_16256/"$ /* if empty then file is in a home dir */ /* if graphic file is empty (= 0 bytes) then run draw2d command again */ draw2d( terminal = 'svg, file_name = sconcat(path,"2_",string(HitLimit)), user_preamble="set size square;set key top right", title= concat("Dynamical plane for fc(z)=z*z+",string(c)), dimensions = [iX_max, iY_max], yrange = [ZyMin,ZyMax], xrange = [ZxMin,ZyMax], xlabel = "Z.re ", ylabel = "Z.im", point_type = filled_circle, points_joined =true, point_size = 0.2, color = red, points_joined =false, color = black, key = "backward orbit of z=beta", points(map(realpart,xy),map(imagpart,xy)), points_joined =false, color = green, point_size = 1.4, key = "critical value", ToPoint(c), key = sconcat("external ray t=",string(t)), color = green, points_joined =true, point_size = 0.2, ToPoints(eRay), points_joined = false, color = black, point_size = 1.4, key = "critical point z = 0.0", ToPoint(0.0), points_joined =true, point_size = 0.2, color = red, key = sconcat("external ray t/2 = ", string(t/2)), ToPoints(eRayT), key = sconcat("external ray (t+1)/2 =",string((t+1)/2)), color = magenta, ToPoints(eRayTp) )$
- ↑ Rational External Rays of the Mandelbrot Set by Dierk Schleicher
- ↑ Hausdorff dimension of biaccessible angles of quadratic Julia sets and of the Mandelbrot set by Henk Bruin joint work with Dierk Schleicher
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current | 15:37, 14 August 2019 | 1,000 × 1,000 (3.58 MB) | Soul windsurfer | User created page with UploadWizard |
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- Fractals/Iterations in the complex plane/def cqp
- Fractals/Iterations in the complex plane/dynamic external rays
- Fractals/Iterations in the complex plane/jlamination
- Fractals/Iterations in the complex plane/misiurewicz
- Fractals/Iterations in the complex plane/p misiurewicz
- Fractals/mandel
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