Dynamic plane

Discrete dynamic in case of complex quadratic polynomial. exterior = white, interior = gray, unknown=red;

How image is changing with various Iteration Max

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: ${\displaystyle J_{f}\,=\partial A_{f}(p)=\partial A_{f}(\infty )}$
• Fatou set ${\displaystyle F_{f}\subset \mathbb {C} }$
  • exterior of Julia set = basin of attraction to infinity: ${\displaystyle A_{f}(\infty )\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{z\in \mathbb {C} :f^{(k)}(z)\to \infty \ as\ k\to \infty \}.}$
  • interior of Julia set = basin of attraction of finite, parabolic fixed point p: ${\displaystyle A_{f}(p)\ {\overset {\underset {\mathrm {def} }{}}{=}}\ \{z\in \mathbb {C} :f^{(k)}(z)\to p\ as\ k\to \infty \}.}$
    • 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] ${\displaystyle K_{f}}$

Ecalle cilinder

Ecalle cylinders^[15] or Ecalle-Voronin cylinders (by Jean Ecalle^[16][17])^[18]

"... 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"^[19]

eggbeater dynamics

• 
  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)^[20][21]

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

Maybe better description of parabolic dynamics will be Hawaiian earrings

parabolic germ

Germ:^[22][23][24]

• 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 ${\displaystyle f(x)=a_{1}x+a_{2}x^{2}+\ldots }$ with ${\displaystyle \vert a_{1}\vert \neq 0,1}$ is holomorphically conjugated to its linear part ${\displaystyle g(x)=a_{1}z}$" (Sylvain Bonnot)^[25]

germ of vector field

The horn map

"the horn map h = Φ ◦ Ψ, where Φ is a shorthand for Φattr and Ψ for Ψrep (extended Fatou coordinate and parameterizations)."^[26]

Petal

repelling petals around fixed point and its preimages

"The petals ... are interesting not only because they give a rather good intuitive idea of the dynamics that arise near a parabolic point, but also because that the dynamic of f0 on a petal can be linearized, i.e., it is conjugated to the shift map T of C defined by T(w) := w + 1." (Laurea Triennale^[27])

There is no unified definition of petals.

Petal of a flower can be:

• attracting / repelling
• small/alfa/big/ (small attracting petals do not ovelap with repelling petals, but big do)

Each petal is invariant under f^period. In other words it is mapped to itself by f^period.

Attracting petal P is a:

• Each petal is invariant under ${\displaystyle f^{n}}$. In other words it is mapped to itself by ${\displaystyle f^{n}}$: ${\displaystyle f^{n}({\overline {P}})\subset P\cup \left\{p\right\}}$
• domain (topological disc) containing:
  • parabolic periodic point p in its boundary ${\displaystyle {\overline {P}}\ni \left\{p\right\}}$ (precisely in its root, which is a common points of all attracting and repelling petals = center of the Lea-Fatou flower)
  • critical point or it's n=period images on the other side (only small ??)
• trap which captures any orbit tending to parabolic point^[28]
• set contained inside component of filled-in Julia set. The attracting petals of parabolic fixed point are contained in it's basin of attraction
• " ... is maximal with respect to this property. This preferred petal P max always has one or more critical points on its boundary."^[29]
• "an attracting petal is the interior of an analytic curve which lies entirely in the Fatou set, has the right tangency property at the fixed point, and is mapped into its interior by the correct power of the map" Scott Sutherland
• "... an attracting petal is a set of points in a sufficient small disk around the periodic point whose forward orbits always remain in the disk under powers of return map." (W P Thurston: On the geometry and dynamics of Iterated rational maps)

Petals ${\displaystyle P_{j}}$ are symmetric with respect to the d-1 directions:

${\displaystyle arg(z)={\frac {2\Pi l}{d-1}}}$

where:

• d is (to do)
• l is from 0 to d-2

Petals can have different size.

If ${\displaystyle \lambda ^{n}=1}$ then Julia set should approach parabolic periodic point in n directions, between n petals.^[30]

"Using the language of holomorphic dynamics, people would say that you are studying the dynamics of a polynomial near the parabolic fixed point ${\displaystyle z=0}$. By a simple linear change of variables, the study of any such parabolic fixed point can be reduced to the study of ${\displaystyle z\mapsto z+z^{2}+O(z^{3})}$. Then you can apply another change ${\displaystyle w=-{\frac {1}{z}}}$. Thus you are reduced to the study of ${\displaystyle f(w)=w+1+O(1/w)}$. If the real part $Re(w)$ is large enough you will obtain ${\displaystyle f^{n}(w)=w+n+O(\log n)}$. This will give you what you want (when going back to the z-variable).

The domain ${\displaystyle Re(w)>R}$ (for large ${\displaystyle R}$) looks like some kind of cardioid (in your particular case) when you visualize it in the z-variable (it's poetically called an attracting petal)." Sylvain Bonnot^[31]

// choose such value that level sets cross at z=0
// choose radius such a
double GivePetalRadius(complex double c, complex double fixed, int n){
	complex double z = 0.0; // critical point
	int k;
	// best for n>1
	int kMax = (n*ChildPeriod)  - 1; // ????
	
	for(k=0;  k<kMax-1; ++k)
		z = z*z + c; // forward iteration
		
	return  cabs(z-fixed)/2.0;


}

Number of petals

In parabolic point child period coincides with parent period

For quadratic polynomials:

Multiplicity = ParentPeriod + ChildPeriod

NumberOfPetals = multiplicity - ParentPeriod

It is because in parabolic case fixed point coincidence with periodic cycle. Length of cycle (child period) is equal to number of petals.

For other polynomial maps:

• 
  One critical point and 5 petals
• 
  13 critical points and 26 petals.
• 
  14 critical points and 14 petals.
• 
  z^4-z has 3 critical points and 6 petals

f(z)	number of petals	explanation
${\displaystyle z^{d}+z}$	d-1	for ${\displaystyle z^{d}+z=z}$ point z=0 has multiplicity d
${\displaystyle z^{d}-z}$	d+2	(?)for ${\displaystyle f^{2}(z)}$ a root z=0 has multiplicity d+3

For f(z)= -z+z^(p+1) parabolic flower has:

• 2p petals for p odd
• p petals for p even^[33]

... (to do)

Sepal

Sepals and petals

Parabolic sepals for internal angle 1 over 1

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."^[34]
• interior of component with parabolic fixed point on it's boundary

Flower

• 
  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 all petals creates a flower with center at parabolic periodic point.^[35]

Cauliflower

Cauliflower or broccoli:^[36]

• 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.

parabolic implosion

Video on YouTube^[37]

Vector field

• 2D vector field and its

Near parabolic fixed point

Orbits near parabolic fixed point and inside Julia set

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

Period 1

One can easily compute boundary point c

${\displaystyle c=c_{x}+c_{y}*i}$

of period 1 hyperbolic component (main cardioid) for given internal angle (rotation number) t using this cpp code by Wolf Jung^[41]

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

or this Maxima 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)
)$

Period 2

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

Periods 1-6

/*

batch file for Maxima CAS 
computing bifurcation points for period 1-6

 Formulae for cycles in the Mandelbrot set II
Stephenson, 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$

Estimation from exterior

Dynamic rays

Parabolic Julia set for internal angle 1 over 15 - made with use of external rays as a aproximation of Julia set near alfa fixed point

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:

period	Inverse iterations	time
1	340	0m0.021s
2	55 573	0m5.517s
3	8 084 815	13m13.800s
4	1 059 839 105	1724m28.990s

C source code - click on the right to view
a.c:
/* c console program to compile: gcc double_t.c -lm -Wall -march=native to run: time ./a.out period EscapeTime time 1 340 0m0.021s 2 55 573 0m5.517s 3 8 084 815 13m13.800s 4 1 059 839 105 1724m28.990s period 1 escape time = 340.000000 internal angle t = 1.000000 period = 1 preperiod = 0 cx = 0.250000 cy = 0.000000 alfax = 0.500000 alfay = 0.000000 real 0m0.021s =============== escape time = 55573.000000 internal angle t = 0.500000 period = 2 preperiod = 0 cx = -0.750000 cy = 0.000000 alfax = -0.500000 alfay = 0.000000 real 0m5.517s =============================== period = 3 c = (-0.125000; 0.649519); alfa = (-0.250000;0.433013) ea = 0.142857; internal angle t = 0.333333 preperiod = 0 for period = 3 escape time = 8084815 real 13m13.800s ===================================== period = 4 c = (0.250000; 0.500000); alfa = (0.000000;0.500000) ea = 0.066667; internal angle t = 0.250000 preperiod = 0 for period = 4 escape time = 1059839105 real 1724m28.990s */ #include <stdio.h> #include <stdlib.h> // malloc #include <math.h> // M_PI; needs -lm also #include <complex.h> unsigned int period = 4;// of child component of Mandelbrot set unsigned int numerator=1 ; unsigned int denominator; double t ; // internal angle of point c of parent component of Mandelbrot set in turns unsigned long int maxiter = 100000; unsigned int preperiod = 0; //of external angle under doubling map double complex z , c, alfa; double ea ;// 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 */ double complex GiveC(double InternalAngleInTurns, double InternalRadius, unsigned int period) { //0 <= InternalRay<= 1 //0 <= InternalAngleInTurns <=1 // period of parent component of Mandelbrot set { 1,2 } double t = InternalAngleInTurns *2*M_PI; // from turns to radians double R2 = InternalRadius * InternalRadius; double Cx, Cy; /* C = Cx+Cy*i */ switch ( period ) { case 1: // main cardioid Cx = (cos(t)*InternalRadius)/2-(cos(2*t)*R2)/4; Cy = (sin(t)*InternalRadius)/2-(sin(2*t)*R2)/4; break; case 2: // only one component Cx = 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 values Cx = 0.0; Cy = 0.0; break; } return Cx+ Cy*I; } /* http://en.wikipedia.org/wiki/Periodic_points_of_complex_quadratic_mappings z^2 + c = z z^2 - z + c = 0 ax^2 +bx + c =0 // ge3neral for of quadratic equation so: a=1 b =-1 c = c so: The discriminant is the d=b^2-4ac d=1-4c = dx+dy*i r(d)=sqrt(dx^2 + dy^2) sqrt(d) = sqrt((r+dx)/2)+-sqrt((r-dx)/2)*i = sx +- sy*i x1=(1+sqrt(d))/2 = beta = (1+sx+sy*i)/2 x2=(1-sqrt(d))/2 = alfa = (1-sx -sy*i)/2 alfa: 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)) double complex GiveAlfaFixedPoint(double complex c) { double dx, dy; //The discriminant is the d=b^2- 4ac = dx+dy*i double r; // r(d)=sqrt(dx^2 + dy^2) double sx, sy; // s = sqrt(d) = sqrt((r+dx)/2)+-sqrt((r-dx)/2)*i = sx + sy*i double ax, ay; // d=1-4c = dx+dy*i dx = 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*i sx = sqrt((r+dx)/2); sy = sqrt((r-dx)/2); // alfa = ax +ay*i = (1-sqrt(d))/2 = (1-sx + sy*i)/2 ax = 0.5 - sx/2.0; ay = sy/2.0; return ax+ay*I; } double DistanceBetween(double complex z1, double complex z2) {double dx,dy; dx = creal(z1) - creal(z2); dy = cimag(z1) - cimag(z2); return sqrt(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)); */ double complex root(double complex z) { double x = creal(z); double y = cimag(z); double u; double v; double r = sqrt(x*x + y*y); v = sqrt(0.5*(r - x)); if (y < 0) v = -v; u = sqrt(0.5*(r + x)); return u + v*I; } double complex preimage(double complex z1, double complex z2, double complex c) { double complex zPrev; zPrev = root(creal(z1) - creal(c) + (cimag(z1) - cimag(c))*I); // choose one of 2 roots if (creal(zPrev)*creal(z2) + cimag(zPrev)*cimag(z2) > 0) return zPrev ; // u+v*i else return -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.html double backray(double t, // external angle in turns int n, //period of ray's angle under doubling map int k, // preperiod int iterMax, double complex c ) { double xend ; // re of the endpoint of the ray double yend; // im of the endpoint of the ray const double R = 4; // very big radius = near infinity int j; // number of ray double iter=0.0; // index of backward iteration double complex zPrev; double u,v; // zPrev = u+v*I double complex zNext; double distance; /* dynamic 1D arrays for coordinates (x, y) of points with the same R on preperiodic and periodic rays */ double *RayXs, *RayYs; int iLength = n+k+2; // length of arrays ?? why +2 // creates arrays: RayXs and RayYs and checks if it was done RayXs = malloc( iLength * sizeof(double) ); RayYs = malloc( iLength * sizeof(double) ); if (RayXs == NULL || RayYs==NULL) { fprintf(stderr,"Could not allocate memory"); getchar(); return 1; } // starting points on preperiodic and periodic rays // with angles t, 2t, 4t... and the same radius R for (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*t if (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 z do { for (j = 0; j < n+k; j++) // period +preperiod { // u+v*i = sqrt(z-c) backward iteration in fc plane zPrev = root(RayXs[j+1] - creal(c)+(RayYs[j+1] - cimag(c))*I); // , u, v u=creal(zPrev); v=cimag(zPrev); // choose one of 2 roots: u+v*i or -u-v*i if (u*RayXs[j] + v*RayYs[j] > 0) { RayXs[j] = u; RayYs[j] = v; } // u+v*i else { 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 0 xend = RayXs[0]; yend = RayYs[0]; // free memmory free(RayXs); free(RayYs); return iter; // } /* ---------------------- main ------------------*/ int main() { double EscapeTime; // internal angle denominator = period; t = (double) numerator/denominator; // c = GiveC(t, 1.0, 1); alfa = GiveAlfaFixedPoint(c); //external angle denominator= 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); // return 0; }

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 ...).

Estimation from interior

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

• find points of interior of Kc
• find boundary of interior of Kc using edge detection

If components of interior are lying very close to each other then find components using:^[44]

color = LastIteration % period

For parabolic components between parent and child component:^[45]

periodOfChild = denominator*periodOfParent  
color = iLastIteration % periodOfChild

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

Attraction time

Various types of dynamics

Interior of filled Julia set consist of components. 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 same attraction 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) return  iColorsOfInterior[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 alpha
i // number of iterations after which critical point z=0.0 reaches disc around fixed point alpha with radius AR
denominator of internal angle (in turns)
internal angle = 1/denominator

Note that attraction time i is proportional to denominator.

Attraction time for various denominators

Now you see what means weakly 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 or fast computer)
• find better method (:-)) if time is to long for you

Trap

Trap = target set

Estimation from interior and exterior

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 repelling points

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

Bof61

• Using bof61 for coloring interior
• example

See also

• Image: Nonstandard Parabolic by Cheritat^[50]
• Julia set of parabolic case in Maxima CAS^[51]
• 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 by Artem 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

References

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. ↑ Théorie des invariants holomorphes. Thèse d'Etat, Orsay, March 1974
16. ↑ Jean Ecalle in french wikipedia
17. ↑ Jean Ecalle home page
18. ↑ 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.
19. ↑ mappings by Luna Lomonaco
20. ↑ MODULUS OF ANALYTIC CLASSIFICATION FOR UNFOLDINGS OF GENERIC PARABOLIC DIFFEOMORPHISMSby P. Mardesic, R. Roussarie¤ and C. Rousseau
21. ↑ mathoverflow questions: the functional equation ffxxfx2
22. ↑ Germ in wikipedia
23. ↑ MODULUS OF ANALYTIC CLASSIFICATION FOR UNFOLDINGS OF GENERIC PARABOLIC DIFFEOMORPHISMS by P. Mardesic , R. Roussarie and C. Rousseau
24. ↑ The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point Colin Christopher, Christiane Rousseau
25. ↑ Mathoverflow: infinitesimal classification of functions near a fixed point upto conjugation
26. ↑ Near parabolic renormalization for unisingular holomorphic maps by Arnaud Cheritat
27. ↑ The Hausdorff dimension of the boundary of the Mandelbrot set. Tesi di Laurea Triennale
28. ↑ PARABOLIC IMPLOSION A MINI-COURSE by ARNAUD CHERITAT
29. ↑ A FAMILY OF DEGREE 4 BLASCHKE PRODUCTS by Jordi Canela
30. ↑ BOF, page 39
31. ↑ Asymptotics of iterated polynomials
32. ↑ An Introduction to Julia and Fatou Sets by Scott Sutherland ( January 2014 DOI10.1007/978-3-319-08105-2__3 )
33. ↑ A Lösungen zu den Übungenn by Michael Becker
34. ↑ Note on dynamically stable perturbations of parabolics by Tomoki Kawahira
35. ↑ wikipedia: Rose (topology)
36. ↑ cauliflower at MuEncy by Robert Munafo
37. ↑ Circle Implodes Into Flames - video by sinflrobot
38. ↑ A Topology Simplification Method For 2D Vector Fields by Xavier Tricoche Gerik Scheuermann Hans Hagen
39. ↑ e ncyclopedia of math: Sector_in_the_theory_of_ordinary_differential_equations
40. ↑ wikipedia: Multiplicity in mathematics
41. ↑ Mandel: software for real and complex dynamics by Wolf Jung
42. ↑ Local dynamics at a fixed point by Evgeny Demidov
43. ↑ Parabolic Julia Sets are Polynomial Time Computable Mark Braverman
44. ↑ The fixed points and periodic orbits by Evgeny Demidov
45. ↑ Src code of c program for drawing parabolic Julia set
46. ↑ stackexchange questions: what-is-the-shape-of-parabolic-critical-orbit
47. ↑ planetmath: San Marco fractal
48. ↑ wikipedia: Douady rabbit
49. ↑ planetmath: San Marco fractal
50. ↑ Image: Nonstandard Parabolic by Cheritat
51. ↑ Julia set of parabolic case in Maxima CAS