"Most programs for computing Julia sets work well when the underlying dynamics is hyperbolic butexperience an exponential slowdown in the parabolic case." (Mark Braverman)^[1]

In other words it means that one can need days for making a good picture of parabolic Julia set with standard / naive algorithms.

There are 2 problems here:

• slow (= lazy) local dynamics (in the neighbourhood of a parabolic fixed point)
• some parts are very thin (hard to find usingstandard plane scanning)

Dynamic plane = complex z-plane${\displaystyle \mathbb{C}=J_f\cup F_f}$:

• Julia set${\displaystyle J_f\subset \mathbb{C}}$ is a common boundary:${\displaystyle J_f=\mathrm{\partial }{A}_f(p)=\mathrm{\partial }{A}_f(\mathrm{\infty })}$
• Fatou set${\displaystyle F_f\subset \mathbb{C}}$
  • exterior of Julia set = basin of attraction to infinity:${\displaystyle A_f(\mathrm{\infty })\stackrel{\underset{\mathrm{d}\mathrm{e}\mathrm{f}}{}}{=}\{z\in \mathbb{C}:f^{(k)}(z)\to \mathrm{\infty }ask\to \mathrm{\infty }\}.}$
  • interior of Julia set = basin of attraction of finite, parabolic fixed point p:${\displaystyle A_f(p)\stackrel{\underset{\mathrm{d}\mathrm{e}\mathrm{f}}{}}{=}\{z\in \mathbb{C}:f^{(k)}(z)\to pask\to \mathrm{\infty }\}.}$
    • immediate basin = sum of components which have parabolic fixed point p on it's boundary  ; the immediate parabolic basin of p is the union of periodic connected components of the parabolic basin.
      • attracting Lea-Fatou flower = sum of n attracting petals = sum of 2*n attracting sepals
        • petal = part of the flower. Each petal contains part of 2 sepals,
        • sepals (Let 1 be an invariant curve in the first quadrant and L 1 the region enclosed by 1 ∪ {0}, called a sepal.)^[2]

${\displaystyle \mathbb{C}\supset F(f)\supset A_f(p)}$

See also:

• Filled Julia set^[3]${\displaystyle K_f}$

• parabolic chessboard or checkerboard
• parabolic implosion
• Fatou coordinate
• Hawaiian earring
  • in wikipedia = "The plane figure formed by a sequence of circles that are all tangent to each other at the same point and such that the sequence of radii converges to zero." (Barile, Margherita in MathWorld)^[4]
  • commons:Category:Hawaiian earrings
• Gevrey symptotic expansions
  • the Écalle-Voronin invariants of (7.1) at the origin which have Gevrey- 1/2 asymptotic expansions^[5]
• germ^[6][7][8]
  • germ of the function: Taylor expansion of the function
• multiplicity^[9]
• Julia-Lavaurs sets
• The Leau-Fatou flower theorem:^[10] repelling or attracting flower. Flower consist of petals
  • Leau-Fatou flower
• Parabolic Linearization theorem
• The horn map
• Blaschke product
• Inou and Shishikura's near parabolic renormalization
• complex polynomial vector field^[11]
• numbers
  • "a positive integer ν,the parabolic degeneracy with the following property: there are νq attracting petals and νq repelling petals, which alternate cyclically around the fixed point."^[12]
  • combinatorial rotation number
• a Poincaré linearizer of function f at parabolic fixed point^[13]
• "the parabolic pencil. This is the family of circles which all have one common point, and thus are all tangent to each other, either internally or externally."^[14]

The Leau-Fatou flower theorem states that, if function${\displaystyle f^n}$ has the Taylor expansion^[15]

${\displaystyle f^n(z)=z+a{z}^{n+1}+O(z^{n+2})}$

then the complex number${\displaystyle v}$ on the unit circle${\displaystyle v\in S^1}$ describes unit vector ( direction):

• an repulsion vector (from origin to v) for f at the origin if${\displaystyle na{v}^n=1}$
• an attraction vector ( from v to origin ) if${\displaystyle na{v}^n=-1}$

There are n equally spaced attracting directions, separated by n equally spaced repelling directions.The integer n+1 is called the multiplicity of fixed point

Function expansion: z+z^5

/* Maxima CAS */display2d:false;taylor(z+z^5, z,0,5);                                z + z^5

So :

• a = 1
• n = 4

Compute attracting directions:

/* Maxima CAS */solve (4*v^4 = 1, v);[v = %i/sqrt(2),v = -1/sqrt(2),v = -%i/sqrt(2),v = 1/sqrt(2)]

Function m*z+z^2

(%i7) taylor(m*z+z^2, z,0,5);(%o7) m*z+z^2

So :

• a = 1
• n = 1

(%i9) solve (v = 1, v);(%o9) [v = 1]

Ecalle cylinders^[16] or Ecalle-Voronin cylinders (by Jean Ecalle^[17][18])^[19]

"... the quotient of a petal P under the equivalence relation identifying z and f (z) if both z and f (z) belong to P. This quotient manifold is called the Ecalle cilinder, and it is conformally isomorphic to the infinite cylinder C/Z"^[20]

• 
  Hand Egg beater
• 
  Here is real model of what happens in parabolic case

Physical model: the behaviour of cake when one uses eggbeater.

The mathematical model: a 2D vector field with 2 centers (second-order degenerate points)^[21][22]

The field is spinning about the centers, but does not appear to be diverging.

Maybe better description of parabolic dynamics will be Hawaiian earrings

Germ:^[23][24][25]

• z+z^2
• z+z^3
• z+z^{k+1}
• z+a_{k+1}z^{k+1}
• z+a_{k+1}z^{k+1}
• "a germ${\displaystyle f(x)=a_{1}x+a_2{x}^2+\dots }$ with${\displaystyle |a_1|\ne 0,1}$ is holomorphically conjugated to its linear part${\displaystyle g(x)=a_{1}z}$" (Sylvain Bonnot)^[26]

germ of vector field

"the horn map h = Φ ◦ Ψ, where Φ is a shorthand for Φattr and Ψ for Ψrep (extended Fatou coordinate and parameterizations)."^[27]

Petal

• "The petals are Jordan domains invariant by${\displaystyle f^q}$" R PEREZ-MARCO^[28]
• Petal is trap which captures any orbit tending to parabolic point
• petal is a part of the flower

Definitions:

• A sepal is the intersection of an attracting and repelling petal.A parabolic Pommerenke-Levin-Yoccoz inequality. by Xavier Buf and Adam L. Epstein There does not seem to be an official definition of sepal.
• "Let l be an invariant curve in the first quadrant and L1 the region enclosed by l ∪ {0}, called a sepal."^[29]
• interior of component with parabolic fixed point on it's boundary

• 
  Lea-Fatu flower
• 
  Flower with four petals and parabolic point in the center
• 
  Critical orbit for internal angle 1/5 showing 5 attracting directions

Sum of allpetals creates a flower^[30] with center at parabolic periodic point.^[31]

Cauliflower or broccoli:^[32]

• empty (its interior is empty) for c outside Mandelbrot set. Julia set is a totally disconnected.
• filled cauliflower for c=1/4 on boundary of the Mandelbrot set. Julia set is a Jordan curve (quasi circle).

• 
  c = 0 < 1/4
• 
  c = 0.25 = 1/4 t he root of the main cardioid. Julia set is a cauliflower.
• 
  c = 0.255 > 1/4 "imploded cauliflower"
• 
  c = 0.258 > 1/4
• 
  c = [0.285, 0.01]

Please note that:

• size of image differs because of different z-planes.
• different algorithms are used so colours are hard to compare.

How Julia set changes along real axis (going from c=0 thru c=1/4 and further):

Perturbation of a function${\displaystyle f(z)}$ by complex${\displaystyle ϵ}$:

${\displaystyle g(z)=f(z)+ϵ}$

When one add epsilon > 0 (move along real axis toward + infinity) there is abifurcation of parabolic fixed point:

• attracting fixed point (epsilon<0)
• one parabolic fixed point (epsilon = 0)
• one parabolic fixed point splits up into two conjugate repelling fixed points (epsilon > 0)

"If we slightly perturb with epsilon<0 then the parabolic fixed point splits up into two real fixed points on the real axis (one attracting, one repelling)."

See:

• demo 2 page 9 in program Mandel by Wolf Jung

Parabolic imposion

• on the parameter plane
  • point c moves from interior of the component, through the boundary to the exterior of Mandelbrot set
  • nucleus ( c=0), along internal ray 0, parabolic point ( c= 0.25), along external ray 0
• on the dynamic plane
  • connected Julia set ( with interior) imploedes and comes disconnected ( without interior)
  • fixed point moves from interior to Julia set ( parabolic)
  • one basin ( interior) disappears

Video on YouTube^[33]

• Parabolic implosion : from circle thru cauliflower to imploded cauliflower
• 
  c=0
• 
  c=1/4
• 
  c= 1/4 + 0.05
• 
  c = 1/4 + 0.029
• 
  c = 1/4 + 0.035

• 2D vector field and its

singularity types:

• center type: "In this case, one can find a neighborhood of the singular point where all integral curves are closed, inside one another, and contain the singular point in their interior"^[34]
• non-center type: neighborhood of singularity is made of several curvilinear sectors:^[35]

"A curvilinear sector is defined as the region bounded by a circle C with arbitrary small radius and two streamlines S and S! both converging towards singularity. One then considers the streamlines passing through the open sector g in order to distinguish between three possible types of curvilinear sectors."

Dynamics:

• global
• local

Local dynamics:

• in the exterior of Julia set
• on the Julia set
• near parabolic fixed point (inside Julia set)

See also

• youtube playlist: ODTWÓRZ WSZYSTKIE Corinna Ulcigrai - Parabolic dynamics and renormalization from Institut des Hautes Études Scientifiques (IHÉS)

Why analyze f^p not f ?

Forward orbit of f near parabolic fixed point is composite. It consist of 2 motions:

• around fixed point
• toward / away from fixed point

• Computing parameters using boundary equations

Type of parabolic parameters:

• root points
• cusps

Parabolic points of period 1 component of Mandelbrot set (parameter plane)

n	Internal angle (rotation number) t = 1/n	The root point c = parabolic parameter	Two external angles of parameter rays landing on the root point c (1/(2^n+1); 2/(2^n+1)	fixed point${\displaystyle z_{\alpha }}$	external angles of dynamic rays landing on fixed point${\displaystyle z_{\alpha }}$
1	1/1	0.25	(0/1 ; 1/1)	0.5	(0/1 = 1/1)
2	1/2	-0.75	(1/3; 2/3)	-0.5	(1/3; 2/3)
3	1/3	0.64951905283833*%i-0.125	(1/7; 2/7)	0.43301270189222*%i-0.25	(1/7; 2/7; 3/7)
4	1/4	0.5*%i+0.25	(1/15; 2/15)	0.5*%i	(1/15; 2/15; 4/15; 8/15)
5	1/5	0.32858194507446*%i+0.35676274578121	(1/31; 2/31)	0.47552825814758*%i+0.15450849718747	(1/31; 2/31; 4/31; 8/31; 16/31)
6	1/6	0.21650635094611*%i+0.375	(1/63; 2/63)	0.43301270189222*%i+0.25	(1/63; 2/63; 4/63; 8/63; 16/63; 32/63)
7	1/7	0.14718376318856*%i+0.36737513441845	(1/127; 2/127)	0.39091574123401*%i+0.31174490092937	(1/127; 2/127, 4/127; 8/127; 16/127; 32/127, 64/127)
8	1/8	0.10355339059327*%i+0.35355339059327		0.35355339059327*%i+0.35355339059327
9	1/9	0.075191866590218*%i+0.33961017714276		0.32139380484327*%i+0.38302222155949
10	1/10	0.056128497072448*%i+0.32725424859374		0.29389262614624*%i+0.40450849718747

For internal angle n/p parabolic period p cycle consist of one z-point with multiplicity p^[36] andmultiplier = 1.0 . This point z is equal to fixed point${\displaystyle z_{alfa}}$

One can easily compute boundary point c

${\displaystyle c=c_x+c_y\ast i}$

of period 1 hyperbolic component (main cardioid) for giveninternal angle (rotation number) t using thiscpp code by Wolf Jung^[37]

t*=(2*PI);// from turns to radianscx=0.5*cos(t)-0.25*cos(2*t);cy=0.5*sin(t)-0.25*sin(2*t);

or thisMaxima CAS code:

 /* conformal map from circle to cardioid ( boundary of period 1 component of Mandelbrot set */F(w):=w/2-w*w/4;/* circle D={w:abs(w)=1 } where w=l(t,r) t is angle in turns ; 1 turn = 360 degree = 2*Pi radians r is a radius */ToCircle(t,r):=r*%e^(%i*t*2*%pi);GiveC(angle,radius):=( [w], /* point of  unit circle   w:l(internalAngle,internalRadius); */ w:ToCircle(angle,radius),  /* point of circle */ float(rectform(F(w)))    /* point on boundary of period 1 component of Mandelbrot set */)$compile(all)$/* ---------- global constants & var ---------------------------*/Numerator: 1;DenominatorMax: 10;InternalRadius: 1;/* --------- main -------------- */for Denominator:1 thru DenominatorMax step 1 do( InternalAngle: Numerator/Denominator, c: GiveC(InternalAngle,InternalRadius), display(Denominator), display(c),  /* compute fixed point */ alfa:float(rectform((1-sqrt(1-4*c))/2)), /* alfa fixed point */ display(alfa))$

// cpp code by W Jung http://www.mndynamics.comt*=(2*PI);// from turns to radianscx=0.25*cos(t)-1.0;cy=0.25*sin(t);

/*batch file for Maxima CAS computing bifurcation points for period 1-6 Formulae for cycles in the Mandelbrot set IIStephenson, John; Ridgway, Douglas T.Physica A, Volume 190, Issue 1-2, p. 104-116.*/kill(all);remvalue(all);start:elapsed_run_time ();/* ------------ functions ----------------------*//* exponential for of complex number with angle in turns */ /* "exponential form prevents allroots from working", code by Robert P. Munafo */GivePoint(Radius,t):=rectform(ev(Radius*%e^(%i*t*2*%pi), numer))$ /* gives point of unit circle for angle t in turns */GiveCirclePoint(t):=rectform(ev(%e^(%i*t*2*%pi), numer))$ /* gives point of unit circle for angle t in turns Radius = 1 *//* gives a list of iMax points of unit circle */GiveCirclePoints(iMax):=block( [circle_angles,CirclePoints], CirclePoints:[], circle_angles:makelist(i/iMax,i,0,iMax), for t in circle_angles do CirclePoints:cons(GivePoint(1,t),CirclePoints), return(CirclePoints) /* multipliers */)$/* http://commons.wikimedia.org/wiki/File:Mandelbrot_set_Components.jpg Boundary equation  b_n(c,P)=0     defines relations between hyperbolic components and unit circle for given period n ,    allows computation of exact coordinates of hyperbolic componenets.b_n(w,c), is boundary polynomial (implicit function of 2 variables).*/GiveBoundaryEq(P,n):=block( if n=1 then return(c + P^2 - P), if n=2 then return(- c + P - 1), if n=3 then return(c^3 + 2*c^2 - (P-1)*c + (P-1)^2), if n=4 then return(c^6 + 3*c^5 + (P+3)* c^4 + (P+3)* c^3 - (P+2)*(P-1)*c^2 - (P-1)^3), if n=5 then return(c^15 + 8*c^14 + 28*c^13 + (P + 60)*c^12 + (7*P + 94)*c^11 +   (3*P^2 + 20*P + 116)*c^10 + (11*P^2 + 33*P + 114)*c^9 + (6*P^2 + 40*P + 94)*c^8 +   (2*P^3 - 20*P^2 + 37*P + 69)*c^7 + (3*P - 11)*(3*P^2 - 3*P - 4)*c^6 + (P - 1)*(3*P^3 + 20*P^2 - 33*P - 26)*c^5 +  (3*P^2 + 27*P + 14)*(P - 1)^2*c^4 - (6*P + 5)*(P - 1)^3*c^3 + (P + 2)*(P - 1)^4*c^2 - c*(P - 1)^5  + (P - 1)^6),if n=6 then return (c^27+13*c^26+78*c^25+(293 - P)*c^24+(792 - 10*P)*c^23+(1672 - 41*P)*c^22+(2892 - 84*P - 4*P^2)*c^21+(4219 - 60*P - 30*P^2)*c^20+(5313 + 155*P - 80*P^2)*c^19+(5892 + 642*P - 57*P^2 + 4*P^3)*c^18+(5843 + 1347*P + 195*P^2 + 22*P^3)*c^17+(5258 + 2036*P + 734*P^2 + 22*P^3)*c^16+(4346 + 2455*P + 1441*P^2 - 112*P^3 + 6*P^4)*c^15 + (3310 + 2522*P + 1941*P^2 - 441*P^3 + 20*P^4)*c^14 + (2331 + 2272*P + 1881*P^2 - 853*P^3 - 15*P^4)*c^13 + (1525 + 1842*P + 1344*P^2 - 1157*P^3 - 124*P^4 - 6*P^5)*c^12 + (927 + 1385*P + 570*P^2 - 1143*P^3 - 189*P^4 - 14*P^5)*c^11 + (536 + 923*P - 126*P^2 - 774*P^3 - 186*P^4 + 11*P^5)*c^10 + (298 + 834*P + 367*P^2 + 45*P^3 - 4*P^4 + 4*P^5)*(1-P)*c^9 + (155 + 445*P - 148*P^2 - 109*P^3 + 103*P^4 + 2*P^5)*(1-P)*c^8 + 2*(38 + 142*P - 37*P^2 - 62*P^3 + 17*P^4)*(1-P)^2*c^7 + (35 + 166*P + 18*P^2 - 75*P^3 - 4*P^4)*((1-P)^3)*c^6 + (17 + 94*P + 62*P^2 + 2*P^3)*((1-P)^4)*c^5 + (7 + 34*P + 8*P^2)*((1-P)^5)*c^4 + (3 + 10*P + P^2)*((1-P)^6)*c^3 + (1 + P)*((1-P)^7)*c^2 +-c*((1-P)^8) + (1-P)^9))$/* gives a list of points c on boundaries on all components for give period */GiveBoundaryPoints(period,Circle_Points):=block( [Boundary,P,eq,roots],  Boundary:[], for m in Circle_Points do (/* map from reference plane to parameter plane */  P:m/2^period,  eq:GiveBoundaryEq(P,period), /* Boundary equation  b_n(c,P)=0  */  roots:bfallroots(%i*eq),  roots:map(rhs,roots),  for root in roots do Boundary:cons(root,Boundary)),  return(Boundary))$/* divide llist of roots to 3 sublists = 3  components *//* ---- boundaries of period 3 components period:3$Boundary3Left:[]$Boundary3Up:[]$Boundary3Down:[]$Radius:1; for m in CirclePoints do (  P:m/2^period,  eq:GiveBoundaryEq(P,period),  roots:bfallroots(%i*eq),  roots:map(rhs,roots),  for root in roots do      (       if realpart(root)<-1  then Boundary3Left:cons(root,Boundary3Left),       if (realpart(root)>-1 and imagpart(root)>0.5)             then Boundary3Up:cons(root,Boundary3Up),       if (realpart(root)>-1 and imagpart(root)<0.5)             then Boundary3Down:cons(root,Boundary3Down)                    ))$--------- *//* gives a list of parabolic points for given: period and internal angle */GiveParabolicPoints(period,t):=block( [m,ParabolicPoints,P,eq,roots], m: GiveCirclePoint(t), /* root of unit circle, Radius=1, angle t=0 */ ParabolicPoints:[], /* map from reference plane to parameter plane */ P:m/2^period, eq:GiveBoundaryEq(P,period), /* Boundary equation  b_n(c,P)=0  */ roots:bfallroots(%i*eq), roots:map(rhs,roots), for root in roots do ParabolicPoints:cons(float(root),ParabolicPoints), return(ParabolicPoints))$compile(all)$/* ------------- constant values ----------------------*/fpprec:16;/* ------------unit circle on a w-plane -----------------------------------------*/a:GiveParabolicPoints(6,1/3);a$

/*gcc c.c -lm -Wall./a.outRoot point between period 1 component and period 987 component  = c = 0.2500101310666710+0.0000000644946597Internal angle (c) = 1/987Internal radius (c) = 1.0000000000000000*/#include<stdio.h>#include<math.h>// M_PI; needs -lm also#include<complex.h>/*   c functions using complex type numbers   computes c from  component  of Mandelbrot set */complexdoubleGive_c(intPeriod,intn,intd,doubleInternalRadius){complexdoublec;complexdoublew;// point of reference plane  where image of the component is a unit disk// alfa = ax +ay*i = (1-sqrt(d))/2 ; // resultdoublet;// InternalAngleInTurnst=(double)n/d;t=t*M_PI*2.0;// from turns to radiansw=InternalRadius*cexp(I*t);// map to the unit diskswitch(Period)// of component{case1:// main cardioid = only one period 1 componentc=w/2-w*w/4;// https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_1break;case2:// only one period 2 componentc=(w-4)/4;// https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set/boundary#Solving_system_of_equation_for_period_2break;// period > 2default:printf("higher periods : to do, use newton method\n");printf("for each q = Period of the Child component  there are 2^(q-1) roots\n");c=10000.0;// bad valuebreak;}returnc;}voidPrintAndDescribe_c(intperiod,intn,intd,doubleInternalRadius){complexdoublec=Give_c(period,n,d,InternalRadius);printf("Root point between period %d component and period %d component  = c = %.16f%+.16f*I\t",period,d,creal(c),cimag(c));printf("Internal angle (c) = %d/%d\n",n,d);//printf("Internal radius (c) = %.16f\n",InternalRadius);}/*https://stackoverflow.com/questions/19738919/gcd-function-for-cThe GCD function uses Euclid's Algorithm.It computes A mod B, then swaps A and B with an XOR swap.*/intgcd(inta,intb){inttemp;while(b!=0){temp=a%b;a=b;b=temp;}returna;}intmain(){intperiod=1;doubleInternalRadius=1.0;// internal angle in turns as a ratio = p/qintn=1;intd=987;// n/d = local angle in turnsfor(n=1;n<d;++n){if(gcd(n,d)==1)// irreducible fraction{PrintAndDescribe_c(period,n,d,InternalRadius);}}return0;}

• 5 methods of drawing
• 
  Checking iteration
• 
  modified DEM
• 
  checking angle
• 
  checking if zn is in target set
• 
  checking if zn is in triangle target set

All points of interior of filled Julia set tend to one periodic orbit (or fixed point). This point is in Julia set and is weakly attracting.^[38] One can analyse only behavior near parabolic fixed point. It can be done usingcritical orbits.

There are two cases here: easy and hard.

If the Julia set near parabolic fixed point is like n-th arm star (not twisted) then one can simply check argument of zn, relative to the fixed point. See for example z+z^5. This is an easy case.

In the hard case Julia set is twisted around fixed.

Julia set of root point is topologically the same as the Julia set of the child period center, but

• thecenter ( nucleus) Julia set is very easy to draw ( superattracting basin = very fast dynamics because critical point is also periodic point)
• while the root Julia set ( parabolic) is hard to draw ( parabolic basin and lasy dynamics)

Examples:

• t = 1/2
  • Julia set of the root point = fat Basilica Julia set: c = -3/4 = - 0.75
  • Julia set of period 2 center = (slim) Basicica Julia set: c = -1
• t = 1/3
  • Julia set of the root point = fatDouady Rabbit: c = -0.125000000000000 +0.649519052838329i
  • Julia set of period 3 center = (slim)Doudy Rabbit Julia set: c = -0.122561166876654 +0.744861766619744i period = 3

Description

Description

• Long iteration method^[39]
• The big steps algorithm^[40]

Steps

• choose fuction${\displaystyle f}$which has fixed parabolic point at origin ( z=0)
• choose internal angle${\displaystyle \theta =\frac{m}{k}}$
• compute${\displaystyle F_k}$ which is an aproximation of higher iterates of function${\displaystyle f}$ for z close to zero using power series centered at zero ( Taylor series = Maclaurin series )
• find how many terms of power series to use and on which annuli to use specific${\displaystyle F_k}$ experimentally

${\displaystyle F_k\approx f^{2^k}}$

each${\displaystyle F_k}$ will be used on an annulus

where K is fixed

Lambda form of complex quadratic polynomial which has an indifferent fixed point with multiplier${\displaystyle \lambda }$ at the origin^[41]

${\displaystyle f_{\lambda }(z)=\lambda z+z^2}$

where:

• multiplier of fixed point${\displaystyle \lambda =e^{2\pi \theta i}}$
• internal angle is an rational number and proper fraction${\displaystyle \theta =\frac{m}{k}}$

Choose

• ${\displaystyle \theta =\frac{1}{7}}$ so${\displaystyle \lambda =e^{(2\pi i)/7}}$
• 30 terms of power series
• approximated function for annuli k=4,5,...,10 and default function f^n for larger values of z ( outside annuli)
• delta for function equal to 10^-5

(* code by Professor: Mark McClure from https://marksmath.org/classes/Spring2019ComplexDynamics/ *)n=7;f[z_]=Exp[2Pi*I/n]z+z^2;Remove[F];F[0][z_]=N[Normal[Series[f[z],{z,0,30}]]];Do[F[0][z_]=Chop[N[Normal[Series[F[0][f[z]],{z,0,30}]]],10^-5],{n-1}];Do[F[k][z_]=Chop[N[Normal[Series[F[k-1][F[k-1][z]],{z,0,30}]]],10^-5],{k,1,10}](* define and compile function FF *)FF=With[{n=n,f4=Function[z,Evaluate[F[4][z]]],f5=Function[z,Evaluate[F[5][z]]],f6=Function[z,Evaluate[F[6][z]]],f7=Function[z,Evaluate[F[7][z]]],f8=Function[z,Evaluate[F[8][z]]],f9=Function[z,Evaluate[F[9][z]]],f10=Function[z,Evaluate[F[10][z]]]},Compile[{{z,_Complex}},Which[Abs[z]>1/2^3,Nest[Function[zz,N[Exp[2Pi*I/n]]zz+zz^2],z,n],Abs[z]<=1/2^9,f10[z],Abs[z]<=1/2^8,f9[z],Abs[z]<=1/2^7,f8[z],Abs[z]<=1/2^6,f7[z],Abs[z]<=1/2^5,f6[z],Abs[z]<=1/2^4,f5[z],Abs[z]<=1/2^3,f4[z],True,0]]];(* iterate 1000 times and then see what happens *)iterate=With[{FF=FF,n=n},Compile[{{z0,_Complex}},Module[{z,i},z=z0;i=0;While[1/2^9<Abs[z]<=2&&i++<1000n,z=FF[z]];z],RuntimeOptions->"Speed",CompilationTarget->"C"]];(* now compute some iteration data *)data=Monitor[Table[iterate[x+I*y],{y,Im[center]+1.2,Im[center],-0.0025},{x,Re[center]-1.2,Re[center]+1.2,0.0025}],y];(* use some symmetry to cut computation time in half *)center=First[Select[z/.NSolve[f[z]==0,z],Im[#]<0&]]/2(* center = -0.311745 - 0.390916*I *)data=Join[data,Reverse[Rest[Reverse/@data]]];(* plot it  *)kernel={{1,1,1},{1,-8,1},{1,1,1}};(*classifyArg=Compile[{{z,_Complex},{z0,_Complex},{v,_Complex},{n,_Integer}},Module[{check,check2},check=n(Arg[(z0-z)/v]+Pi)/(2Pi);check2=Ceiling[check];If[check==check2,0,check2]]];classified=Map[classify,data,{2}];convolvedData=ListConvolve[kernel,classified];ArrayPlot[Sign[Abs[convolvedData]]]

Mathematical Functions of Wolfram language (  :

• Series[f,{x,x0,n}] generates a power series expansion for f about the point${\displaystyle x=x_0}$ to order${\displaystyle (x-x_0{)}^n}$, where n is an explicit integer
• N[expr] gives the numerical value of expr
• Chop[expr,delta] replaces numbers smaller in absolute magnitude than delta by 0
• Normal[expr] converts expr to a normal expression from a variety of special forms.
• Do[expr,n] evaluates expr n times
• Do[expr,{i,imin,imax}] evaluates expr and starts with i=imin
• With[{x=x0,y=y0,…},expr] specifies that all occurrences of the symbols x, y, … in expr should be replaced by x0, y0, ….
• Compile[{{x1,t1},…},expr] assumes that xi is of a type that matches ti.
• Which[test1,value1,test2,value2,…] evaluates each of the testi in turn, returning the value of the valuei corresponding to the first one that yields True
• Nest[f,expr,n] gives an expression with f applied n times to expr.

One can use periodic dynamic rays landing on parabolic fixed point to find narrow parts of exterior.

Let's check how many backward iterations needs point on periodic ray with external radius = 4 to reach distance 0.003 from parabolic fixed point:

/*c console programto compile: gcc double_t.c -lm -Wall -march=nativeto run:time ./a.outperiodEscapeTimetime1       340 0m0.021s2    55 573    0m5.517s3 8 084 81513m13.800s4    1 059 839 105    1724m28.990speriod 1escape time   = 340.000000internal angle t  = 1.000000period  = 1preperiod  = 0cx  = 0.250000cy  = 0.000000alfax  = 0.500000alfay  = 0.000000real0m0.021s===============escape time   = 55573.000000internal angle t  = 0.500000period  = 2preperiod  = 0cx  = -0.750000cy  = 0.000000alfax  = -0.500000alfay  = 0.000000real0m5.517s===============================period = 3  c = (-0.125000; 0.649519); alfa  = (-0.250000;0.433013)ea  = 0.142857;internal angle t  = 0.333333preperiod  = 0for period = 3  escape time =    8084815real13m13.800s=====================================period = 4  c = (0.250000; 0.500000); alfa  = (0.000000;0.500000)ea  = 0.066667;internal angle t  = 0.250000preperiod  = 0for period = 4  escape time = 1059839105real1724m28.990s*/#include<stdio.h>#include<stdlib.h> // malloc#include<math.h> // M_PI; needs -lm also#include<complex.h>unsignedintperiod=4;// of child component of Mandelbrot setunsignedintnumerator=1;unsignedintdenominator;doublet;// internal angle of point c of parent component of Mandelbrot set in turnsunsignedlongintmaxiter=100000;unsignedintpreperiod=0;//of external angle under doubling mapdoublecomplexz,c,alfa;doubleea;// External Angle/* find c in component of Mandelbrot set uses complex type so #include <complex.h> and -lm uses code by Wolf Jung from program Mandel see function mndlbrot::bifurcate from mandelbrot.cpp http://www.mndynamics.com/indexp.html  */doublecomplexGiveC(doubleInternalAngleInTurns,doubleInternalRadius,unsignedintperiod){//0 <= InternalRay<= 1//0 <= InternalAngleInTurns <=1// period of parent component of Mandelbrot set  { 1,2 }doublet=InternalAngleInTurns*2*M_PI;// from turns to radiansdoubleR2=InternalRadius*InternalRadius;doubleCx,Cy;/* C = Cx+Cy*i */switch(period){case1:// main cardioidCx=(cos(t)*InternalRadius)/2-(cos(2*t)*R2)/4;Cy=(sin(t)*InternalRadius)/2-(sin(2*t)*R2)/4;break;case2:// only one componentCx=InternalRadius*0.25*cos(t)-1.0;Cy=InternalRadius*0.25*sin(t);break;// for each period  there are 2^(period-1) roots.default:// safe valuesCx=0.0;Cy=0.0;break;}returnCx+Cy*I;}/*http://en.wikipedia.org/wiki/Periodic_points_of_complex_quadratic_mappingsz^2 + c = zz^2 - z + c = 0ax^2 +bx + c =0 // ge3neral for  of quadratic equationso:a=1b =-1c = cso:The discriminant is the d=b^2-4acd=1-4c = dx+dy*ir(d)=sqrt(dx^2 + dy^2)sqrt(d) = sqrt((r+dx)/2)+-sqrt((r-dx)/2)*i = sx +- sy*ix1=(1+sqrt(d))/2 = beta = (1+sx+sy*i)/2x2=(1-sqrt(d))/2 = alfa = (1-sx -sy*i)/2alfa: attracting when c is in main cardioid of Mandelbrot set, then it is in interior of Filled-in Julia set, it means belongs to Fatou set (strictly to basin of attraction of finite fixed point)*/// uses global variables://  ax, ay (output = alfa(c))doublecomplexGiveAlfaFixedPoint(doublecomplexc){doubledx,dy;//The discriminant is the  d=b^2- 4ac = dx+dy*idoubler;// r(d)=sqrt(dx^2 + dy^2)doublesx,sy;// s = sqrt(d) = sqrt((r+dx)/2)+-sqrt((r-dx)/2)*i = sx + sy*idoubleax,ay;// d=1-4c = dx+dy*idx=1-4*creal(c);dy=-4*cimag(c);// r(d)=sqrt(dx^2 + dy^2)r=sqrt(dx*dx+dy*dy);//sqrt(d) = s =sx +sy*isx=sqrt((r+dx)/2);sy=sqrt((r-dx)/2);// alfa = ax +ay*i = (1-sqrt(d))/2 = (1-sx + sy*i)/2ax=0.5-sx/2.0;ay=sy/2.0;returnax+ay*I;}doubleDistanceBetween(doublecomplexz1,doublecomplexz2){doubledx,dy;dx=creal(z1)-creal(z2);dy=cimag(z1)-cimag(z2);returnsqrt(dx*dx+dy*dy);}/* principal square  root of complex number http://en.wikipedia.org/wiki/Square_root z1= I;  z2 = root(z1);  printf("zx  = %f \n", creal(z2));  printf("zy  = %f \n", cimag(z2));*/doublecomplexroot(doublecomplexz){doublex=creal(z);doubley=cimag(z);doubleu;doublev;doubler=sqrt(x*x+y*y);v=sqrt(0.5*(r-x));if(y<0)v=-v;u=sqrt(0.5*(r+x));returnu+v*I;}doublecomplexpreimage(doublecomplexz1,doublecomplexz2,doublecomplexc){doublecomplexzPrev;zPrev=root(creal(z1)-creal(c)+(cimag(z1)-cimag(c))*I);// choose one of 2 rootsif(creal(zPrev)*creal(z2)+cimag(zPrev)*cimag(z2)>0)returnzPrev;//  u+v*ielsereturn-zPrev;// -u-v*i}// This function only works for periodic or preperiodic angles.// You must determine the period n and the preperiod k before calling this function.// based on same function from src code of program Mandel by Wolf Jung// http://www.mndynamics.com/indexp.htmldoublebackray(doublet,// external angle in turnsintn,//period of ray's angle under doubling mapintk,// preperiodintiterMax,doublecomplexc){doublexend;// re of the endpoint of the raydoubleyend;// im of the endpoint of the rayconstdoubleR=4;// very big radius = near infinityintj;// number of raydoubleiter=0.0;// index of backward iterationdoublecomplexzPrev;doubleu,v;// zPrev = u+v*IdoublecomplexzNext;doubledistance;/* dynamic 1D arrays for coordinates (x, y) of points with the same R on preperiodic and periodic rays  */double*RayXs,*RayYs;intiLength=n+k+2;// length of arrays ?? why +2//  creates arrays:  RayXs and RayYs  and checks if it was doneRayXs=malloc(iLength*sizeof(double));RayYs=malloc(iLength*sizeof(double));if(RayXs==NULL||RayYs==NULL){fprintf(stderr,"Could not allocate memory");getchar();return1;}//  starting points on preperiodic and periodic rays//  with angles t, 2t, 4t...  and the same radius Rfor(j=0;j<n+k;j++){// z= R*exp(2*Pi*t)RayXs[j]=R*cos((2*M_PI)*t);RayYs[j]=R*sin((2*M_PI)*t);t*=2;// t = 2*tif(t>1)t--;// t = t modulo 1}zNext=RayXs[0]+RayYs[0]*I;//printf("RayXs[0]  = %f \n", RayXs[0]);//printf("RayYs[0]  = %f \n", RayYs[0]);// z[k] is n-periodic. So it can be defined here explicitly as well.RayXs[n+k]=RayXs[k];RayYs[n+k]=RayYs[k];//   backward iteration of each point zdo{for(j=0;j<n+k;j++)// period +preperiod{// u+v*i = sqrt(z-c)   backward iteration in fc planezPrev=root(RayXs[j+1]-creal(c)+(RayYs[j+1]-cimag(c))*I);// , u, vu=creal(zPrev);v=cimag(zPrev);// choose one of 2 roots: u+v*i or -u-v*iif(u*RayXs[j]+v*RayYs[j]>0){RayXs[j]=u;RayYs[j]=v;}// u+v*ielse{RayXs[j]=-u;RayYs[j]=-v;}// -u-v*i}// for j ...//RayYs[n+k] cannot be constructed as a preimage of RayYs[n+k+1]RayXs[n+k]=RayXs[k];RayYs[n+k]=RayYs[k];// convert to pixel coordinates//  if z  is in window then draw a line from (I,K) to (u,v) = part of ray// printf("for iter = %d cabs(z) = %f \n", iter, cabs(RayXs[0] + RayYs[0]*I));iter+=1.0;distance=DistanceBetween(RayXs[j]+RayYs[j]*I,alfa);printf("distance  = %10.9f ; iter = %10.0f\n",distance,iter);// info}while(distance>0.003);// distance < pixel size// last point of a ray 0xend=RayXs[0];yend=RayYs[0];// free memmoryfree(RayXs);free(RayYs);returniter;//}/* ---------------------- main ------------------*/intmain(){doubleEscapeTime;// internal angledenominator=period;t=(double)numerator/denominator;//c=GiveC(t,1.0,1);alfa=GiveAlfaFixedPoint(c);//external angledenominator=pow(2,period)-1;ea=(double)1.0/denominator;//EscapeTime=backray(ea,period,preperiod,maxiter,c);//printf("period = %d ",period);printf(" c = (%f; %f);",creal(c),cimag(c));printf(" alfa  = (%f;%f)\n",creal(alfa),cimag(alfa));printf("ea  = %f;\n",ea);printf("internal angle t  = %f\n",t);printf("preperiod  = %d\n",preperiod);printf("for period = %d  escape time = %10.0f\n",period,EscapeTime);//return0;}

One can use only argument of point z of external rays and its distance to alfa fixed point (see code from image). It works for periods up to 15 (maybe more ...).

Julia set is a boundary of filled-in Julia set Kc.

• find points of interior of Kc
• find boundary of interior of Kc usingedge detection

If components of interior are lying very close to each other then find components using:^[42]

color = LastIteration % period

For parabolic components between parent and child component:^[43]

periodOfChild = denominator*periodOfParent  color = iLastIteration % periodOfChild

where denominator is a denominator of internal angle of parent component of Mandelbrot set.

"if the iterate zn of tends to a fixed parabolic point, then the initial seed z0 is classified according to the argument of zn−z0, the classification being provided by the flower theorem" (Mark McClure^[44])

Interior of filled Julia set consist ofcomponents. All comonents are preperiodic, some of them are periodic (immediate basin of attraction).

In other words:

• one iteration moves z to another component (and whole component to another component)
• all point of components have the sameattraction time (number of iteration needed to reach target set around attractor)

It is possible to use it to color components. Because in the parabolic case the attractor is weak (weakly attracting) it needs a lot of iterations for some points to reach it.

// i = number of iteration// iPeriodChild = period of child component of Mandelbrot set ( parabolic c value is a root point between parant and child component/* distance from z to Alpha  */Zxt=Zx-dAlfaX;Zyt=Zy-dAlfaY;d2=Zxt*Zxt+Zyt*Zyt;// interior: check if fall into internal target set (circle around alfa fixed point)if(d2<dMaxDistance2Alfa2)returniColorsOfInterior[i%iPeriodChild];

Here are some example values:

 iWidth  = 1001 // width of image in pixels PixelWidth  = 0.003996   AR  = 0.003996 // Radius around attractor denominator  = 1 ; Cx  = 0.250000000000000; Cy  = 0.000000000000000 ax  = 0.500000000000000; ay  = 0.000000000000000    denominator  = 2 ; Cx  = -0.750000000000000; Cy  = 0.000000000000000 ax  = -0.500000000000000; ay  = 0.000000000000000    denominator  = 3 ; Cx  = -0.125000000000000; Cy  = 0.649519052838329 ax  = -0.250000000000000; ay  = 0.433012701892219   denominator  = 4 ; Cx  = 0.250000000000000; Cy  = 0.500000000000000 ax  = 0.000000000000000; ay  = 0.500000000000000    denominator  = 5 ; Cx  = 0.356762745781211; Cy  = 0.328581945074458 ax  = 0.154508497187474; ay  = 0.475528258147577    denominator  = 6 ; Cx  = 0.375000000000000; Cy  = 0.216506350946110 ax  = 0.250000000000000; ay  = 0.433012701892219      denominator  = 1 ;   i =               243.000000  denominator  = 2 ;   i =            31 171.000000  denominator  = 3 ;   i =         3 400 099.000000  denominator  = 4 ;   i =       333 293 206.000000  denominator  = 5 ;   i =    29 519 565 177.000000  denominator  = 6 ;   i = 2 384 557 783 634.000000

where:

C = Cx + Cy*i a = ax + ay*i // fixed point alphai // number of iterations after which critical point z=0.0 reaches disc around fixed point alpha with radius ARdenominator of internal angle (in turns)internal angle = 1/denominator

Note that attraction time i is proportional to denominator.

Now you see what meansweakly attracting.

One can:

• use brutal force method (Attracting radius < pixelSize; iteration Max big enough to let all points from interior reach target set; long time orfast computer)
• find better method (:-)) if time is to long for you

Julia set is a common boundary of filled-in Julia set and basin of attraction of infinity.

• find points of interior/components of Kc
• find escaping points
• find boundary points using Sobel filter

It works for denominator up to 4.

Inverse iteration of alfa fixed point. It works good only for cutting point (where external rays land). Other points still are not hitten.

• Using bof61 for coloring interior
• example

• Critical orbits in case of one critical point
• 
  critical orbits for internal angle from 1/1 to 1/10. Trueattracting directions
• 
  N-th arm stars for n from 1 to 10.Schematic attracting directions
• 
  critical orbit, attracting and repelling vectors for internal angle 1/3

• from period 1 thru ...
• 
  1/1cauliflower
• 
  1/2San Marco fractal^[45]
• 
  1/3 Douady fat rabbit^[46]
• 
  1/4
• 
  1/5
• 
  1/7
• 
  1/10
• 
  1/15
• 
  1/20
• 
  1/30

• from period 2 thru ...
• 
  from 2 thru internal ray1/1 ; c=-0.75 Julia set is aSan Marco fractal^[47]
• 
  from period 2 thru1/2
• 
  from period 2 thru1/3
• 
  from period 2 thru1/4

• from period 3 thru ...
• 
  from period 3 thru1/3

External examples:

• A parabolic point with 377 petalsparabolic renormalization of z^3+c by YANG Fei
• A quad. Julia sets with 987 and 89 petals by A Cheritat

Forother polynomial maps see here

• Image: Nonstandard Parabolic by Cheritat^[48]
• Julia set of parabolic case in Maxima CAS^[49]
• The parabolic Mandelbrot set Pascale ROESCH (joint work with C. L. PETERSEN
• PARABOLIC IMPLOSION A MINI-COURSE by ArnaudCheritat
• Workshop on parabolic implosion 2010
• The renormalization for parabolic fixed points... Mitsushiro Shishikura and Hiroyuki Inou
• Dynamics of holomorphic maps: Resurgence of Fatou coordinates, and Poly-time computability of Julia sets byArtem Dudko,arxiv.org: Dudko_A
• Minicourse "Analytic classification of germs of generic families unfolding a parabolic point by Christiane Rousseau
• Perspectives on Parabolic Points in Holomorphic Dynamics - The Banff International Research Station for Mathematical Innovation and Discovery (BIRS)
• Richard Oudkerk: The Parabolic Implosion for f_0(z) = z + z^{nu+1} + O(z^{nu+2})
• Perspectives on Parabolic Points in Holomorphic Dynamics Videos from BIRS Workshop 15w5082
• Generic Perturbation of Parabolic Points Having More Than One Attracting Petal by Arnaud Chéritat
• Parabolic fixed points - Mathematica Notebook by Mark McClure

1. ↑Mark Braverman: On efficient computation of parabolic Julia sets
2. ↑Note on dynamically stable perturbations of parabolics by Tomoki Kawahira
3. ↑Filled Julia set in wikipedia
4. ↑Barile, Margherita. "Hawaiian Earring." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/HawaiianEarring.html
5. ↑Augustin Fruchard, Reinhard Sch¨afke. Composite Asymptotic Expansions and Difference Equations. Revue Africaine de la Recherche en Informatique et Math´ematiques Appliqu´ees, INRIA, 2015, 20, pp.63-93. <hal-01320625>
6. ↑wikipedia: Germ(mathematics)
7. ↑Fixed points of diffeomorphisms, singularities of vector fields and epsilon-neighborhoods of their orbits by Maja Resman
8. ↑The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point by Colin Christopher, Christiane Rousseau
9. ↑wikipedia: Multiplicity (mathematics)
10. ↑Dynamics of surface homeomorphisms Topological versions of the Leau-Fatou flower theorem and the stable manifold theorem by Le Roux, F
11. ↑The Dynamics of Complex Polynomial Vector Fields in C by Kealey Dias
12. ↑LIMITS OF DEGENERATE PARABOLIC QUADRATIC RATIONAL MAPS by XAVIER BUFF, JEAN ECALLE, AND ADAM EPSTEIN
13. ↑Poincaré linearizers in higher dimensionsby Alastair Fletcher
14. ↑pencil of circles by James King
15. ↑math.stackexchange question: what-is-the-shape-of-parabolic-critical-orbit
16. ↑Théorie des invariants holomorphes. Thèse d'Etat, Orsay, March 1974
17. ↑Jean Ecalle in french wikipedia
18. ↑Jean Ecalle home page
19. ↑Lukas Geyer - Normal forms via uniformization (10/28/2016). This is the draft of the proof of local normal forms at attracting, repelling, and parabolic fixed points using the uniformization theorem, handed out in class. It will eventually be incorporated into the lecture notes.
20. ↑mappings by Luna Lomonaco
21. ↑MODULUS OF ANALYTIC CLASSIFICATION FOR UNFOLDINGS OF GENERIC PARABOLIC DIFFEOMORPHISMSby P. Mardesic, R. Roussarie¤ and C. Rousseau
22. ↑mathoverflow questions: the functional equation ffxxfx2
23. ↑Germ in wikipedia
24. ↑MODULUS OF ANALYTIC CLASSIFICATION FOR UNFOLDINGS OF GENERIC PARABOLIC DIFFEOMORPHISMS by P. Mardesic , R. Roussarie and C. Rousseau
25. ↑The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point Colin Christopher, Christiane Rousseau
26. ↑Mathoverflow: infinitesimal classification of functions near a fixed point upto conjugation
27. ↑Near parabolic renormalization for unisingular holomorphic maps by Arnaud Cheritat
28. ↑Ricardo Pérez-Marco. "Fixed points and circle maps." Acta Math. 179 (2) 243 - 294, 1997. https://doi.org/10.1007/BF02392745
29. ↑Note on dynamically stable perturbations of parabolics by Tomoki Kawahira
30. ↑Parabolic fixed points: The Leau-Fatou Flower by Davide Legacci March 18, 2021
31. ↑wikipedia: Rose (topology)
32. ↑cauliflower at MuEncy by Robert Munafo
33. ↑Circle Implodes Into Flames - video by sinflrobot
34. ↑A Topology Simplification Method For 2D Vector Fields by Xavier Tricoche Gerik Scheuermann Hans Hagen
35. ↑encyclopedia of math: Sector_in_the_theory_of_ordinary_differential_equations
36. ↑wikipedia: Multiplicity in mathematics
37. ↑Mandel: software for real and complex dynamics by Wolf Jung
38. ↑Local dynamics at a fixed point by Evgeny Demidov
39. ↑Parabolic Julia Sets are Polynomial Time Computable Mark Braverman
40. ↑Mark McClure class Spring2019: ComplexDynamics , see code "Iteration near parabolic points" ( Mathematica notebook )
41. ↑Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
42. ↑The fixed points and periodic orbits by Evgeny Demidov
43. ↑Src code of c program for drawing parabolic Julia set
44. ↑stackexchange questions: what-is-the-shape-of-parabolic-critical-orbit
45. ↑planetmath: San Marco fractal
46. ↑wikipedia: Douady rabbit
47. ↑planetmath: San Marco fractal
48. ↑Image: Nonstandard Parabolic by Cheritat
49. ↑Julia set of parabolic case in Maxima CAS

• Book:Fractals