Fractals/Iterations in the complex plane/def cqp

< Fractals





Internal addresses describe the combinatorial structure of the Mandelbrot set.[1]

Internal address :

  • is not constant within hyperbolic component. Example : internal address of -1 is 1->2 and internal address of 0.9999 is 1[2]
  • of hyperbolic component is defined as a internal address of it's center


Angled internal address is an extension of internal address


Types of angleEdit

Principal branch or complex number argument
external angle internal angle plain angle
parameter plane  arg(\Phi_M(c))  \,  arg(\rho_n(c)) \,  arg(c) \,
dynamic plane  arg(\Phi_c(z)) \,  arg(z) \,

where :


The external angle is a angle of point of set's exterior. It is the same on all points on the external ray


The internal angle is an angle of point of component's interior

  • it is a rational number and proper fraction measured in turns
  • it is the same for all point on the internal ray
  • in a contact point ( root point ) it agrees with the rotation number
  • root point has internal angle 0

\alpha = \frac{p}{q} \in \mathbb{Q}


The plain angle is an agle of complex point = it's argument [3]


  • turns
  • degrees
  • radians

Number typesEdit

Angle ( for example external angle in turns ) can be used in different number types

Examples :

the external arguments of the rays landing at z = −0.15255 + 1.03294i are :[4]

(\theta^- _{20} , \theta^+_{20} ) = (0.\overline{00110011001100110100}, 0.\overline{00110011001101000011})

where :

\theta^- _{20}  = 0.\overline{00110011001100110100}_2 = 0.\overline{20000095367522590181913549340772}_{10} = \frac{209716}{1048575} = \frac{209716}{2^{20}-1}


Coordinate :



  • topology: closed versus open
  • other properities:
    • invariant
    • critical


Closed curves are curves whose ends are joined. Closed curves do not have end points.

  • Simple Closed Curve : A connected curve that does not cross itself and ends at the same point where it begins. It divides the plane into exactly two regions ( Jordan curve theorem ). Examples of simple closed curves are ellipse, circle and polygons.[5]
  • complex Closed Curve ( not simple = non-simple ) It divides the plane into more than two regions. Example : Lemniscates.

"non-self-intersecting continuous closed curve in plane" = "image of a continuous injective function from the circle to the plane"


Inner circleEdit

Unit circleEdit

Unit circle \partial D\, is a boundary of unit disk[6]

\partial D = \left\{ w: abs(w)=1  \right \}

where coordinates of w\, point of unit circle in exponential form are :

w = e^{i*t}\,

Critical curvesEdit

Diagrams of critical polynomials are called critical curves.[7]

These curves create skeleton of bifurcation diagram.[8] (the dark lines[9])

Escape linesEdit

"If the escape radius is equal to 2 the contour lines have a contact point (c= -2) and cannot be considered as equipotential lines" [10]


curve is invariant for the map f ( evolution function ) if images of every point from the curve stay on that curve


Equipotential linesEdit

Equipotential lines = Isocurves of complex potential

"If the escape radius is greater than 2 the contour lines are equipotential lines" [11]

Jordan curveEdit

Illustration of the Jordan curve theorem. The Jordan curve (drawn in black) divides the plane into an "inside" region (light blue) and an "outside" region (pink).

Jordan curve = a simple closed curve that divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points[12]


Lamination of the unit disk is a closed collection of chords in the unit disc, which can intersect only in an endpoint of each on the boundary circle[13][14]

It is a model of Mandelbrot or Julia set.

A lamination, L, is a union of leaves and the unit circle which satisfies :[15]

  • leaves do not cross (although they may share endpoints) and
  • L is a closed set.


Chords = leaves = arcs

A leaf on the unit disc is a path connecting two points on the unit circle. [16]

Open curveEdit

Curve which is not closed. Examples : line, ray.


Rays are :

  • invariant curves
  • dynamic or parameter
  • external or internal

External rayEdit

Internal rayEdit

Dynamic internal ( blue segment) and external ( red ray) rays

Internal rays are :

  • dynamic ( on dynamic plane , inside filled Julia set )
  • parameter ( on parameter plane , inside Mandelbrot set )


A spider S is a collection of disjoint simple curves called legs [17]( extended rays = external + internal ray) in the complex plane connecting each of the post-critical points to infnity [18]

See :


"A vein in the Mandelbrot set is a continuous, injective arc inside in the Mandelbrot set"

"The principal vein v_{p/q} is the vein joining c_{p/q} to the main cardioid" (Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems. A dissertation by Giulio Tiozzo )



Derivative of Iterated function (map)

Derivative with respect to cEdit

On parameter plane :

  • c is a variable
  • z_0 = 0 is constant
\frac{d}{dc} f^{(p)} _c (z_0) = z'_p \,

This derivative can be found by iteration starting with

z_0 = 0 \,
z'_0 = 1 \,

and then

z_p = z_{p-1}^2 + c \,
z'_p = 2 \cdot z_{p-1}\cdot z'_{p-1} + 1 \,

This can be verified by using the chain rule for the derivative.

  • Maxima CAS function :

dcfn(p, z, c) :=
  if p=0 then 1
  else 2*fn(p-1,z,c)*dcfn(p-1, z, c)+1;

Example values :

z_0 = 0 \qquad\qquad z'_0 = 1 \,
z_1 = c \qquad\qquad z'_1 =  1 \,
z_2 = c^2+c \qquad z'_2 = 2c+1 \,

Derivative with respect to zEdit

z'_n\, is first derivative with respect to c.

This derivative can be found by iteration starting with

z'_0 = 1 \,

and then :

z'_n= 2*z_{n-1}*z'_{n-1}\,


Germ [19] of the function f in the neighborhood of point z is a set of the functions g which are indistinguishable in that neighborhood

[f]_z = \{g : g \sim_z f\}.

See :


  • Iterated function = map[20]
  • an evolution function[21] of the discrete nonlinear dynamical system[22]
z_{n+1} = f_c(z_n)  \,

is called map f_c :

f_c : z \to z^2 + c. \,


  • The map f is hyperbolic if every critical orbit converges to a periodic orbit.[23]

Complex quadratic mapEdit


c form : z^2+cEdit

quadratic map[24]

  • math notation : f_c(z)=z^2+c\,
  • Maxima CAS function :
(%i1) z:zx+zy*%i;
(%o1) %i*zy+zx
(%i2) c:cx+cy*%i;
(%o2) %i*cy+cx
(%i3) f:z^2+c;
(%o3) (%i*zy+zx)^2+%i*cy+cx
(%i4) realpart(f);
(%o4) -zy^2+zx^2+cx
(%i5) imagpart(f);
(%o5) 2*zx*zy+cy

Iterated quadratic map

  • math notation

 \ f^{(0)} _c (z) =   z = z_0
 \ f^{(1)} _c (z) =   f_c(z) = z_1


 \ f^{(p)} _c (z) =   f_c(f^{(p-1)} _c (z))

or with subscripts :

 \ z_p =  f^{(p)} _c (z_0)
  • Maxima CAS function :
fn(p, z, c) :=
  if p=0 then z
  elseif p=1 then f(z,c)
  else f(fn(p-1, z, c),c);
zp:fn(p, z, c);
lambda form :  z^2+\lambda zEdit

More description Maxima CAS code ( here m not lambda is used )  :

(%i2) z:zx+zy*%i;
(%o2) %i*zy+zx
(%i3) m:mx+my*%i;
(%o3) %i*my+mx
(%i4) f:m*z+z^2;
(%o4) (%i*zy+zx)^2+(%i*my+mx)*(%i*zy+zx)
(%i5) realpart(f);
(%o5) -zy^2-my*zy+zx^2+mx*zx
(%i6) imagpart(f);
(%o6) 2*zx*zy+mx*zy+my*zx
Switching between formsEdit

Start from :

  • internal angle \theta = \frac {p}{q}
  • internal radius r

Multiplier of fixed point :

\lambda = r e^{2 \pi \theta i}

When one wants change from lambda to c :[25]

c = c(\lambda) = \frac {\lambda}{2} \left(1 - \frac {\lambda}{2}\right) = \frac {\lambda}{2} - \frac {\lambda^2}{4}

or from c to lambda :

\lambda = \lambda(c) = 1 \pm  \sqrt{1- 4 c}

Example values :

\theta r c fixed point alfa z_c \lambda fixed point z_{\lambda}
1/1 1.0 0.25 0.5 1.0 0
1/2 1.0 -0.75 -0.5 -1.0 0
1/3 1.0 0.64951905283833*i-0.125 0.43301270189222*i-0.25 0.86602540378444*i-0.5 0
1/4 1.0 0.5*i+0.25 0.5*i i 0
1/5 1.0 0.32858194507446*i+0.35676274578121 0.47552825814758*i+0.15450849718747 0.95105651629515*i+0.30901699437495 0
1/6 1.0 0.21650635094611*i+0.375 0.43301270189222*i+0.25 0.86602540378444*i+0.5 0
1/7 1.0 0.14718376318856*i+0.36737513441845 0.39091574123401*i+0.31174490092937 0.78183148246803*i+0.62348980185873 0
1/8 1.0 0.10355339059327*i+0.35355339059327 0.35355339059327*i+0.35355339059327 0.70710678118655*i+0.70710678118655 0
1/9 1.0 0.075191866590218*i+0.33961017714276 0.32139380484327*i+0.38302222155949 0.64278760968654*i+0.76604444311898 0
1/10 1.0 0.056128497072448*i+0.32725424859374 0.29389262614624*i+0.40450849718747 0.58778525229247*i+0.80901699437495

One can easily compute parameter c as a point c inside main cardioid of Mandelbrot set :

 c = c_x + c_y*i

of period 1 hyperbolic component ( main cardioid) for given internal angle ( rotation number) t using this c / cpp code by Wolf Jung[26]

double InternalAngleInTurns;
double InternalRadius;
double t = InternalAngleInTurns *2*M_PI; // from turns to radians
double R2 = InternalRadius * InternalRadius;
double Cx, Cy; /* C = Cx+Cy*i */
// main cardioid
Cx = (cos(t)*InternalRadius)/2-(cos(2*t)*R2)/4; 
Cy = (sin(t)*InternalRadius)/2-(sin(2*t)*R2)/4; 

or this Maxima CAS code :

/* conformal map  from circle to cardioid ( boundary
 of period 1 component of Mandelbrot set */

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 

 /* 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 */


/* ---------- global constants & var ---------------------------*/
Numerator :1;
DenominatorMax :10;

/* --------- main -------------- */
for Denominator:1 thru DenominatorMax step 1 do
 InternalAngle: Numerator/Denominator,
 c: GiveC(InternalAngle,InternalRadius),
  /* compute fixed point */
 alfa:float(rectform((1-sqrt(1-4*c))/2)), /* alfa fixed point */

Circle mapEdit

Circle map [27]

  • irrational rotation[28]

Doubling mapEdit

definition [29]

C function ( using GMP library) :

// rop = (2*op ) mod 1 
void mpq_doubling(mpq_t rop, const mpq_t op)
  mpz_t n; // numerator
  mpz_t d; // denominator
  mpz_inits(n, d, NULL);

  mpq_get_num (n, op); // 
  mpq_get_den (d, op); 
  // n = (n * 2 ) % d
  mpz_mul_ui(n, n, 2); 
  mpz_mod( n, n, d);
  // output
  mpq_set_num(rop, n);
  mpq_set_den(rop, d);
  mpz_clears(n, d, NULL);

  • Maxima CAS function using numerator and denominator as an input
doubling_map(n,d):=mod(2*n,d)/d $

or using rational number as an input


  • Common Lisp function
(defun doubling-map (ratio-angle)
" period doubling map =  The dyadic transformation (also known as the dyadic map, 
 bit shift map, 2x mod 1 map, Bernoulli map, doubling map or sawtooth map "
(let* ((n (numerator ratio-angle))
       (d (denominator ratio-angle)))
  (setq n  (mod (* n 2) d)) ; (2 * n) modulo d
  (/ n d))) ; result  = n/d
  • Haskell function[30]
-- by Claude Heiland-Allen
-- type Q = Rational
 double :: Q -> Q
 double p
   | q >= 1 = q - 1
   | otherwise = q
   where q = 2 * p
  • C++
//  mndcombi.cpp  by Wolf Jung (C) 2010. 
// n is a numerator
// d is a denominator
// f = n/d is a rational fraction ( angle in turns )
// twice is doubling map = (2*f) mod 1
// n and d are changed ( Arguments passed to function by reference)

void twice(unsigned long long int &n, unsigned long long int &d)
{  if (n >= d) return;
   if (!(d & 1)) { d >>= 1; if (n >= d) n -= d; return; }
   unsigned long long int large = 1LL; 
   large <<= 63; //avoid overflow:
   if (n < large) { n <<= 1; if (n >= d) n -= d; return; }
   n -= large; 
   n <<= 1; 
   large -= (d - large); 
   n += large;

Inverse function of doubling mapEdit

Every angle α ∈ R/Z measured in turns has :

In Maxima CAS :

InvDoublingMap(r):= [r/2, (r+1)/2];

Note that difference between these 2 preimages

\frac{\alpha}{2} - \frac{\alpha +1}{2} = \frac{1}{2}

is half a turn = 180 degrees = Pi radians.

Images and preimages under doubling map d
\alpha d^1(\alpha) d^{-1}(\alpha)
\frac{1}{2} \frac{1}{1} \left \{ \frac{1}{4} , \frac{3}{4} \right \}
\frac{1}{3} \frac{2}{3} \left \{ \frac{1}{6} , \frac{4}{6} \right \}
\frac{1}{4} \frac{1}{2} \left \{ \frac{1}{8} , \frac{5}{8} \right \}
\frac{1}{5} \frac{2}{5} \left \{ \frac{1}{10} , \frac{6}{10} \right \}
\frac{1}{6} \frac{1}{3} \left \{ \frac{1}{12} , \frac{7}{12} \right \}
\frac{1}{7} \frac{2}{7} \left \{ \frac{1}{14} , \frac{4}{7} \right \}

First return mapEdit

definition [32]

"In contrast to a phase portrait, the return map is a discrete description of the underlying dynamics. .... A return map (plot) is generated by plotting one return value of the time series against the previous one "[33]

"If x is a periodic point of period p for f and U is a neighborhood of x, the composition f^{\circ p}\, maps U to another neighborhood V of x. This locally defined map is the return map for x." ( W P Thurston : On the geometry and dynamics of Iterated rational maps)

"The first return map S → S is the map defined by sending each x0 ∈ S to the point of S where the orbit of x0 under the system first returns to S." [34]

"way to obtain a discrete time system from a continuous time system, called the method of Poincar´e sections Poincar´e sections take us from : continuous time dynamical systems on (n + 1)-dimensional spaces to discrete time dynamical systems on n-dimensional spaces"[35]

Multiplier mapEdit

Multiplier map  \lambda gives an explicit uniformization of hyperbolic component \Eta by the unit disk \mathbb{D}  :

 \lambda : \Eta \to \mathbb{D}

Multiplier map is a conformal isomorphism.[36]

Rotation mapEdit




Critical polynomial :

Q_n = f_c^n(z_{cr}) = f_c^n(0) \,


Q_1 = f_c^1(0) = c \,

Q_2 = f_c^2(0) = c^2 + c \,

Q_3 = f_c^3(0) = (c^2 + c)^2 + c \,

These polynomials are used for finding :

  • centers of period n Mandelbrot set components. Centers are roots of n-th critical polynomials centers = \{ c : f_c^n(z_{cr}) = 0 \}\, ( points where critical curve Qn croses x axis )
  • Misiurewicz points M_{n,k} = \{ c : f_c^k(z_{cr}) = f_c^{k+n}(z_{cr}) \}\,

post-critically finiteEdit

a post-critically finite polynomial = all critical points have finite orbit


glitches = Incorrect parts of renders[37] using perturbation techique


  • periodic points
    • fixed point
  • invariant curve
    • periodic ray
      • external
      • internal


a partition of an interval into subintervals

  • Markov paritition[38]



S(x) is an itinerary of point x under the map f is a right-infinite sequence of zeros and ones [39]

  S(x) = s_1s_2s_3...s_n


Examples :

For a unimodal map with a critical point x_c and invariant interval I :

 I = [0,1] 

one can split interval into 2 subintervals :

 I_0 =[0,x_c) I_0 =(x_c, 1]

then compute s according to it's relation with critical point :

 s_n = \begin{cases}
0 : x_n < x_c \\
1 : x_n > x_c

Itinerary can be converted[40] to point x \in [0, 1]

 \gamma(S_n) = 0.s_1s_2s_3...s_n = \sum_{n=0}^{n-1}\frac{s_n}{2^n} = x_n


magnitude of the point = it's distance from the origin


Multiplier of periodic z-point : [41]

Math notation :

\lambda_c(z) = \frac{df_c^{(p)}(z)}{dz}\,

Maxima CAS function for computing multiplier of periodic cycle :


where p is a period. It takes period as an input, not z point.

period f^p(z) \, \lambda_c(z) \,
1 z^2 + c \, 2z \,
2 z^4 + 2cz^2 + c^2 + c 4z^3 + 4cz
3 z^8 + 4cz^6 + 6c^2z^4 + 2cz^4 + 4c^3z^2 + 4c^2z^2 + c^4 + 2c^3 + c^2 + c 8z^7 + 24cz^5 + 24c^2z^3 + 8cz^3 + 8c^3z + 8c^2z

It is used to :

  • compute stability index of periodic orbit ( periodic point) = |\lambda| = r ( where r is a n internal radius
  • multiplier map


Rotation numberEdit

The rotation number[42][43][44] of the disk ( component) attached to the main cardioid of the Mandelbrot set is a proper, positive rational number p/q in lowest terms where :

  • q is a period of attached disk ( child period ) = the period of the attractive cycles of the Julia sets in the attached disk
  • p descibes fc action on the cycle : fc turns clockwise around z0 jumping, in each iteration, p points of the cycle [45]

Features :

  • in a contact point ( root point ) it agrees with the internal angle
  • the rotation numbers are ordered clockwise along the boundary of the componant
  • " For parameters c in the p/q-limb, the filled Julia set Kc has q components at the fixed point αc . These are permuted cyclically by the quadratic polynomial fc(z), going p steps counterclockwise " Wolf Jung

Winding numberEdit



Orbit is a sequence of points = trajectory



Forward orbit[47] of a critical point[48][49] is called a critical orbit. Critical orbits are very important because every attracting periodic orbit[50] attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.[51][52] [53]

z_0 = z_{cr} = 0\,

z_1 = f_c(z_0) = c\,

z_2 = f_c(z_1) = c^2 +c\,

z_3 = f_c(z_2) = (c^2 + c)^2 + c\,

... \,

This orbit falls into an attracting periodic cycle.

Here are images of critical orbits[54]


Homoclinic / heteroclinicEdit


Inverse = Backward


Parameter ( point of parameter plane ) " is renormalizable if restriction of some of its iterate gives a polinomial-like map of the same or lower degree. " [55]


The smallest positive integer value p for which this equality

 f^p(z_0) = z_0 

holds is the period[56] of the orbit.[57]

 z_0 is a point of periodic orbit ( limit cycle ) \{z_0, \dots , z_{p-1} \}.

More is here


  • Perturbation technque for fast rendering the deep zoom images of the Mandelbrot set[58]
  • perturbation of parabolic point [59]


Planes [60]

Douady’s principle : “sow in dynamical plane and reap in parameter space”.

Dynamic planeEdit

  • z-plane for fc(z)= z^2 + c
  • z-plane for fm(z)= z^2 + m*z

Parameter planeEdit

See :[61]

Types of the parameter plane :

  • c-plane ( standard plane )
  • exponential plane ( map) [62][63]
  • flatten' the cardiod ( unroll ) [64][65] = "A region along the cardioid is continuously blown up and stretched out, so that the respective segment of the cardioid becomes a line segment. .." ( Figure 4.22 on pages 204-205 of The Science Of Fractal Images)[66]
  • transformations [67]



the band-merging points are Misiurewicz points[68]


If there exist two distinct external rays landing at point we say that it is a biaccessible point. [69]


Nucleus or center of hyperbolic componentEdit

A center of a hyperbolic component H is a parameter  c_0 \in H\, ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." [70]

Synonyms :

  • Nucleus of a Mu-Atom [71]

How to find center/s ?

Center of Siegel DiscEdit

Center of Siegel disc is a irrationally indifferent periodic point.

Mane's theorem :

"... appart from its center, a Siegel disk cannot contain any periodic point, critical point, nor any iterated preimage of a critical or periodic point. On the other hand it can contain an iterated image of a critical point." [72]


A critical point[73] of f_c\, is a point  z_{cr} \, in the dynamical plane such that the derivative vanishes:

f_c'(z_{cr}) = 0. \,


f_c'(z) = \frac{d}{dz}f_c(z) = 2z


 z_{cr} = 0\,

we see that the only (finite) critical point of f_c \, is the point  z_{cr} = 0\,.

z_0 is an initial point for Mandelbrot set iteration.[74]


The "neck" of this eight-like figure is a cut-point.
Cut points in the San Marco Basilica Julia set. Biaccessible points = landing points for 2 external rays

Cut point k of set S is a point for which set S-k is dissconected ( consist of 2 or more sets).[75] This name is used in a topology.

Examples :

  • root points of Mandelbrot set
  • Misiurewicz points of boundary of Mandelbrot set
  • cut points of Julia sets ( in case of Siegel disc critical point is a cut point )

These points are landing points of 2 or more external rays.

Point which is a landing point of 2 external rays is called biaccesible

Cut ray is a ray which converges to landing point of another ray. [76] Cut rays can be used to construct puzzles.

Cut angle is an angle of cut ray.


Periodic point when period = 1


Self similarity in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-x direction. The display center pans from (−1, 0) to (−1.31, 0) while the view magnifies from 0.5 × 0.5 to 0.12 × 0.12 to approximate the Feigenbaum ratio \delta.

The Feigenbaum Point[77] is a :

  • point c of parameter plane
  • is the limit of the period doubling cascade of bifurcations
  • an infinitely renormalizable parameter of bounded type
  • boundary point between chaotic ( -2 < c < MF ) and periodic region ( MF< c < 1/4)[78]

MF^{(n)} (\tfrac{p}{q}) = c

Generalized Feigenbaum points are :

  • the limit of the period-q cascade of bifurcations
  • landing points of parameter ray or rays with irrational angles

Examples :

  • MF^{(0)} = MF^{(1)} (\tfrac{1}{2}) = c = -1.401155
  • -.1528+1.0397i)

The Mandelbrot set is conjectured to be self- similar around generalized Feigenbaum points[79] when the magnification increases by 4.6692 (the Feigenbaum Constant) and period is doubled each time[80]

n Period = 2^n Bifurcation parameter = cn Ratio = \dfrac{c_{n-1}-c_{n-2}}{c_n-c_{n-1}}
1 2 -0.75 N/A
2 4 -1.25 N/A
3 8 -1.3680989 4.2337
4 16 -1.3940462 4.5515
5 32 -1.3996312 4.6458
6 64 -1.4008287 4.6639
7 128 -1.4010853 4.6682
8 256 -1.4011402 4.6689
9 512 -1.401151982029
10 1024 -1.401154502237
infinity -1.4011551890 ...

Bifurcation parameter is a root point of period = 2^n component. This series converges to the Feigenbaum point c = −1.401155

The ratio in the last column converges to the first Feigenbaum constant.

" a "Feigenbaum point" (an infinitely renormalizable parameter of bounded type, such as the famous Feigenbaum value which is the limit of the period-2 cascade of bifurcations), then Milnor's hairiness conjecture, proved by Lyubich, states that rescalings of the Mandelbrot set converge to the entire complex plane. So there is certainly a lot of thickness near such a point, although again this may not be what you are looking for. It may also prove computationally intensive to produce accurate pictures near such points, because the usual algorithms will end up doing the maximum number of iterations for almost all points in the picture." Lasse Rempe-Gillen[81]


The point at infinity [82]" is a superattracting fixed point, but more importantly its immediate basin of attraction - that is, the component of the basin containing the fixed point itself - is completely invariant (invariant under forward and backwards iteration). This is the case for all polynomials (of degree at least two), and is one of the reasons that studying polynomials is easier than studying general rational maps (where e.g. the Julia set - where the dynamics is chaotic - may in fact be the whole Riemann sphere). The basin of infinity supports foliations into "external rays" and "equipotentials", and this allows one to study the Julia set. This idea was introduced by Douady and Hubbard, and is the basis of the famous "Yoccoz puzzle"." Lasse Rempe-Gillen[83]


Misiurewicz point[84] = " parameters where the critical orbit is pre-periodic.

Examples are:

  • band-merging points of chaotic bands (the separator of the chaotic bands Bi−1 and Bi )[85]
  • the branch points
  • tips in the Mandelbrot set ( tips of the midgets ) [86]

Characteristic Misiurewicz pointof the chaotic band of the Mandelbrot set is :[87]

  • the most prominent and visible Misiurewicz point of a chaotic band
  • have the same period as the band
  • have the same period as the gene of the band


MF = the Myrberg-Feigenbaum point is the different name for the Feigenbaum Point.

Parabolic pointEdit

parabolic points : this occurs when two singular points coallesce in a double singular point (parabolic point)[88]


Point z has period p under f if :

 z : \ f^{p} (z) =   z


"Pinching points are found as the common landing points of external rays, with exactly one ray landing between two consecutive branches. They are used to cut M or K into well-defined components, and to build topological models for these sets in a combinatorial way. " ( definition from Wolf Jung program Mandel )

See for examples :

  • period 2 = Mandel, demo 2 page 3.
  • period 3 = Mandel, demo 2 page 5 [89]


A post-critical point is a point

 z = f(f(f( ... (z_{cr})))) 

where z_{cr} is a critical point. [90]


precritical points, i.e., the preimages of 0


The root point :

  • has a rotational number 0
  • it is a biaccesible point ( landing point of 2 external rays )


orbit portraitEdit


There are two types of orbit portraits: primitive and satellite. [91]If v is the valence of an orbit portrait \mathcal P and r is the recurrent ray period, then these two types may be characterized as follows:

  • Primitive orbit portraits have r = 1 and v = 2. Every ray in the portrait is mapped to itself by f^n. Each A_j is a pair of angles, each in a distinct orbit of the doubling map. In this case, r_{\mathcal P} is the base point of a baby Mandelbrot set in parameter space.
  • Satellite ( non-primitive ) orbit portraits have r = v \ge 2. In this case, all of the angles make up a single orbit under the doubling map. Additionally, r_{\mathcal P} is the base point of a parabolic bifurcation in parameter space.

Processes and phenomenonaEdit

Contraction and dilatationEdit

  • the contraction z → z/2
  • the dilatation z → 2z.

Implosion and explosionEdit

Explosion (above) and implosion ( below)

Implosion is :

  • the process of sudden change of quality fuatures of the object, like collapsing (or being squeezed in)
  • the opposite of explosion

Example : parabolic implosion in complex dynamics, when filled Julia for complex quadratic polynomial set looses all its interior ( when c goes from 0 along internal ray 0 thru parabolic point c=1/4 and along extrnal ray 0 = when c goes from interior , crosses the bounday to the exterior of Mandelbrot set)[92]

Explosion is a :

  • is a sudden change of quality fuatures of the object in an extreme manner,
  • the opposite of implosion

Example : in exponential dynamics when λ> 1/e , the Julia set of E_{\lambda}(z) = \lambda e^z is the entire plane.[93]




Conformal radiusEdit

Conformal radius of Siegel Disk [95][96]

Escape radius ( ER)Edit

Escape radius ( ER ) or bailout value is a radius of circle target set used in bailout test

Minimal Escape Radius should be grater or equal to 2 :

 ER  =  max ( 2 , |c| )\,

Better estimation is :[97][98]

ER = \frac{1}{2} +\sqrt{\frac{1}{4} + |c| }

Inner radiusEdit

Inner radius of Siegel Disc

  • radius of inner circle, where inner circle with center at fixed point is the biggest circle inside Siegel Disc.
  • minimal distance between center of Siel Disc and critical orbit

Internal radiusEdit

Internal radius is a:


A sequence is an ordered list of objects (or events).[99]

A series is the sum of the terms of a sequence of numbers.[100] Some times these names are not used as in above definitions.


Orbit is a sequence of points





Components of parameter planeEdit

mu-atom , ball, bud, bulb, decoration, lake and lakelet.[102]


Names :

  • mini Mandelbrot set
  • 'baby'-Mandelbrot set
  • island mu-molecules = embedded copy of the Mandelbrot Set[103]
  • Bug
  • Island
  • Mandelbrotie
  • Midget

List of islands :

Primary and satelliteEdit

"Hyperbolic components come in two kinds, primitive and satellite, depending on the local properties of their roots." [104]

Child (Descendant ) and the parentEdit

def [105]

Hyperbolic component of Mandelbrot setEdit

Boundaries of hyperbolic components of Mandelbrot set

Domain is an open connected subset of a complex plane.

"A hyperbolic component H of Mandelbrot set is a maximal domain (of parameter plane) on which f_c\, has an attracting periodic orbit.

A center of a H is a parameter  c_0 \in H\, ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." [106]

A hyperbolic component is narrow if it contains no component of equal or lesser period in its wake [107]


13/34 limb and wake on the left image

p/q limb is a part of Mandelbrot set contained inside p/q wake


Wakes of Mandelbrot Set to Period 10

Wake is the region of parameter plane enclosed by two external rays landing on the same root point on the boundary of main cardioid ( period 1 hyperbolic component).

Angles of the external rays that land on the root point one can find by :

Components of dynamical planeEdit

In case of Siegel disc critical orbit is a boundary of component containing Siegel Disc.


Domain in mathematical analysis it is an open connected set

Jordan domainEdit

"A Jordan domain[108] J is the the homeomorphic image of a closed disk in E2. The image of the boundary circle is a Jordan curve, which by the Jordan Curve Theorem separates the plane into two open domains, one bounded, the other not, such that the curve is the boundary of each." [109]


Lea-Fatu flower

Planar setEdit

a non-separating planar set is a set whose complement in the plane is connected.[110]



Target setEdit

How target set is changing along internal ray 0

Elliptic caseEdit

Target set in elliptic case = inner circle

For the elliptic dynamics, when there is a Siegel disc, the target set is an inner circle

Hyperbolic caseEdit

Infinity is allways hyperbolic attractor for forward iteration of polynomials. Target set here is an exterior of any shape containing all point of Julia set ( and it's interior). There are also other hyperbolic attractors.

In case of forward iteration target set T\, is an arbitrary set on dynamical plane containing infinity and not containing points of filled Julia set.

For escape time algorithms target set determines the shape of level sets and curves. It does not do it for other methods.

Exterior of circleEdit

This is typical target set. It is exterior of circle with center at origin z = 0 \, and radius =ER :

T_{ER}=\{z:abs(z) > ER \} \,

Radius is named escape radius ( ER ) or bailout value.

Circle of radius=ER centered at the origin is :  \{z:abs(z) = ER \} \,

Exterior of squareEdit

Here target set is exterior of square of side length s\, centered at origin

T_s=\{z: abs(re(z)) > s  ~~\mbox{or}~~  abs(im(z))>s \} \,

Parabolic case : petalEdit

trap in parabolic case

In the parabolic case target set shoul be iside petal


Trap is an another name of the target set. It is a set which captures any orbit tending to point inside the trap ( fixed / periodic point ).


Bailout test or escaping testEdit

Two sets after bailout test: escaping white and non-escaping black
Distance to fixed point for various types of dynamics

It is used to check if point z on dynamical plane is escaping to infinity or not.[111] It allows to find 2 sets :

  • escaping points ( it should be also the whole basing of attraction to infinity)[112]
  • not escaping points ( it should be the complement of basing of attraction to infinity)

In practice for given IterationMax and Escape Radius :

  • some pixels from set of not escaping points may contain points that escape after more iterations then IterationMax ( increase IterMax )
  • some pixels from escaping set may contain points from thin filaments not choosed by maping from integer to world ( use DEM )

If z_n is in the target set T\, then z_0 is escaping to infinity ( bailouts ) after n forward iterations ( steps).[113]

The output of test can be :

  • boolean ( yes/no)
  • integer : integer number (value of the last iteration)

Attraction testEdit


Hubbard treeEdit

"Hubbard trees are finite planar trees, equipped with self-maps, which classify postcritically finite polynomials as holomorphic dynamical systems on the complex plane." [114]


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