Fractals/Iterations in the complex plane/def cqp

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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


This address describes period 6 component which is a satelite of period 3 component.


Types of angle[edit]

Principal branch or complex number argument
external angle internal angle plain angle
parameter plane
dynamic plane

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[3] 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
  • "The internal angles start at 0, at the cusp, and increase counterclockwise. " Robert Munafo[4]


The plain angle is an agle of complex point = it's argument [5]


  • turns
  • degrees
  • radians

Number types[edit]

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 :[6]

where :


  • Numerical Bifurcation Analysis of Maps


Coordinate :

   "The coordinates are the current location, measured on the x-y-z axis. The gradient is a direction to move from our current location" Sadid Hasan[8]



  • topology:
    • closed versus open
    • simple versus not simple
  • other properities:
    • invariant
    • critical


  • plane curve = it lies in a plane.
  • closed = it starts and ends at the same place.
  • simple = it never crosses itself.


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.[10]
  • 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 circle[edit]

Unit circle[edit]

Unit circle is a boundary of unit disk[11]

where coordinates of point of unit circle in exponential form are :

Critical curves[edit]

Diagrams of critical polynomials are called critical curves.[12]

These curves create skeleton of bifurcation diagram.[13] (the dark lines[14])


  • a locally connected branched curve
  • "Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic."[15]
  • "a dendrite is a locally connected continuum that does not contain Jordan curves." [16]
  • "a locally connected continuum without subsets homeomorphic to a circle"

Escape lines[edit]

"If the escape radius is equal to 2 the contour lines have a contact point (c= -2) and cannot be considered as equipotential lines" [17]



  • topological
  • shift invariants

examples :


Equipotential lines[edit]

Equipotential lines = Isocurves of complex potential

"If the escape radius is greater than 2 the contour lines are equipotential lines" [22]

Jordan curve[edit]

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[23]


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[24][25]

It is a model of Mandelbrot or Julia set.

A lamination, L, is a union of leaves and the unit circle which satisfies :[26]

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

"The pattern of rays landing together can be described by a lamination of the disk. As θ is varied, the diameter defined by θ/2 and (θ +1)/2 is moving and disconnecting or reconnecting chords. " Wolf Jung [27]


Chords = leaves = arcs

A leaf on the unit disc is a path connecting two points on the unit circle.[28]

Open curve[edit]

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


Rays are :

  • invariant curves
  • dynamic or parameter
  • external, internal or extended


"We prolong an external ray R θ supporting a Fatou component U (ω) up to its center ω through an internal ray and call the resulting set the extended ray E θ with argument θ." Alfredo Poirier[29]

External ray[edit]

Internal ray[edit]

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


  • "The internal rays are the preimages of the radial segments under the coordinate with componenet center corresponding to 0." Alfredo Poirier[30]

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 [31]( extended rays = external + internal ray) in the complex plane connecting each of the post-critical points to infnity [32]

See :


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

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


In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them.



"Symbolic dynamics encodes :




differential equations

  • exact analytic solutions.
  • approximated solution
    • use perturbation theory to approximate the solutions


Field is a region in space where each and every point is associated with a value.

The field types according to the value type :

  • scalar field
  • vector field, for example gradient field



Derivative with respect to c[edit]

On parameter plane :

  • is a variable
  • is constant

This derivative can be found by iteration starting with

and then ( compute derivative before next z because it uses previous values of z):

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 :

It can be used for:

  • the distance estimation method for drawing a Mandelbrot set ( DEM/M )

Derivative with respect to z[edit]

is first derivative with respect to z.

This derivative can be found by iteration starting with

and then :

It can be used for :

  • computing the external distance to the Julia set ( DEM/J )
  • detection of interior points of Mandelbrot set componnets[42]

The Schwarzian Derivative[edit]

The Schwarzian Derivative [43] [44][45][46]

Wirtinger derivatives[edit]


the gradient is the generalization of the derivative for the multivariable functions[47][48]


  • (field): Gradient field is the vector field with gradient vector
  • (function): The gradient of a scalar-valued multivariable function is a vector-valued function denoted
  • (vector): The gradient of the function f at the point (x,y) is defined as the unique vector ( result of gradient function) representing the maximum rate of increase of a scalar function ( length of the vector) and the direction of this maximal rate ( angle of the vector). Such vector is given by the partial derivatives with respect to each of the independent variables[49]
  • (operator): Del or nabla is an gradient operator = a vector differential operator


See also

  • Gradient Descent Algorithm[50][51]
  • Gradient Ascent Algorithm
  • image gradient


The Jacobian is the generalization of the gradient for vector-valued functions of several variables


Germ [52] of the function f in the neighborhood of point z is a set of the functions g which are indistinguishable in that neighborhood

See :


  • differences between map and the function [53]
  • Iterated function = map[54]
  • an evolution function[55] of the discrete nonlinear dynamical system[56]

is called map  :

types or names[edit]



meromorphic maps: Those with NO FINITE, NON-ATTRACTING FIXED POINTS[57]



Critical polynomial :


These polynomials are used for finding :

  • centers of period n Mandelbrot set components. Centers are roots of n-th critical polynomials ( points where critical curve Qn croses x axis )
  • Misiurewicz points

post-critically finite[edit]

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


"resurgent functions display at each of their singular points a behaviour closely related to their behaviour at the origin. Loosely speaking, these functions resurrect, or surge up - in a slightly different guise, as it were - at their singularities"

J. Écalle, 1980[58]

Yoccoz’s function[edit]



  • Incorrect parts of renders[59] using perturbation techique
  • pixels which dynamics differ significantly from the dynamics of the reference pixel[60]"These can be detected and corrected by using a more appropriate reference."[61]


Dessin d'enfant[edit]

See also:


Farey tree[edit]

Farey tree = Farey sequence as a tree

Hubbard tree[edit]

  • "Hubbard trees are finite planar trees, equipped with self-maps, which classify postcritically finite polynomials as holomorphic dynamical systems on the complex plane." [62]
  • " Hubbard trees are invariant trees connecting the points of the critical orbits of post-critically finite polynomials. Douady and Hubbard showed in the Orsay Notes that they encode all combinatorial properties of the Julia sets. For quadratic polynomials, one can describe the

dynamics as a subshift on two symbols, and itinerary of the critical value is called the kneading sequence." Henk Bruin and Dierk Schleicher[63]




  • magnitude of the point ( complex number in 2D case) = it's distance from the origin[64]
  • radius is the absolute value of complex number ( compare to arguments or angle)




  • The map f is hyperbolic if every critical orbit converges to a periodic orbit.[65]

Complex quadratic map[edit]


c form : [edit]

quadratic map[66]

  • math notation :
  • 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


or with subscripts :

  • 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 : [edit]

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 forms[edit]

Start from :

  • internal angle
  • internal radius r

Multiplier of fixed point :

When one wants change from lambda to c :[67]

or from c to lambda :

Example values :

r c fixed point alfa fixed point
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 :

of period 1 hyperbolic component ( main cardioid) for given internal angle ( rotation number) t using this c / cpp code by Wolf Jung[68]

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 map[edit]

Circle map [69]

  • irrational rotation[70]

Doubling map[edit]

definition [71][72]

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[73]
-- 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 map[edit]

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

is half a turn = 180 degrees = Pi radians.

Images and preimages under doubling map d

Feigenbaum map[edit]

"the Feigenbaum map F is a solution of Cvitanovic-Feigenbaum equation"[75]

First return map[edit]

"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 "[77]

"If x is a periodic point of period p for f and U is a neighborhood of x, the composition 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." [78]

"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"[79]

Multiplier map[edit]

Mandelbrot set - multiplier map

Multiplier map associated with hyperbolic component

  • gives an explicit uniformization of hyperbolic component by the unit disk  :

In other words it maps hyperbolic component H to unit disk D.

It maps point c from parameter plane to point b from reference plane:


  • c is a point in the parameter plane
  • b is a point in the reference plane. It is also internal coordinate
  • is a multiplier map

Multiplier map is a conformal isomorphism.[80]

It can be computed using :

Riemann map[edit]

Riemann mapping theorem[81] says that every simply connected subset U of the complex number plane can be mapped to the open unit disk D


  • D is a unit disk
  • f is Riemann map ( function)
  • U is subset of complex plane


  • multiplier map on the parameter plane
  • Böttcher coordinates
    • on the parameter plane the Riemann map for the complement of the Mandelbrot set
    • on dynamic plane[82]
      • for the Fatou component containing a superattracting fixed point for a rational map[83]
      • a Riemann map for the complement of the filled Julia set of a quadratic polynomial with connected Julia : "The Riemann map for the central component for the Basilica was drawn in essentially the same way, except that instead of starting with points on a big circle, I started with sample points on a circle of small radius (e.g. 0.00001) around the origin." Jim Belk


  • explicit formula ( only in simple cases)
  • numerical aproximation ( in most of the cases)[84]
    • Zipper
    • " Thurston and others have done some beautiful work involving approximating arbitrary Riemann maps using circle packings. See Circle Packing: A Mathematical Tale by Stephenson."
    • " To some extent, constructing a Riemann map is simply a matter of constructing a harmonic function on a given domain (as well as the associated harmonic conjugate), subject to certain boundary conditions. The solution to such problems is a huge topic of research in the study of PDE's, although the connection with Riemann maps is rarely mentioned." Jim Belk[85]

PDE's approach to construct a Riemann map explicitly on a given domain D

  • First, translate the domain so that it contains the origin.
  • Next, use a numerical method to construct a harmonic function F satisfying

for all , and let



  • and is harmonic


  • R is the radial component (i.e. modulus) of a Riemann map on D.
  • The angular component can now be determined by the fact that its level curves are perpendicular to the level curves of R, and have equal angular spacing near the origin."

See commons : Category:Riemann_mapping

Rotation map[edit]

     "If a is rational, then every point is periodic. If a is irrational, then every point has a dense orbit." David Richeson[86]


Rotation map describes counterclockwise rotation of point thru turns on the unit circle :


It is used for computing :

Shift map[edit]

names :

  • bit shift map ( because it shifts the bit ) = if the value of an iterate is written in binary notation, the next iterate is obtained by shifting the binary point one bit to the right, and if the bit to the left of the new binary point is a "one", replacing it with a zero.
  • 2x mod 1 map ( because it is math description of it's action )

Shift map (one-sided binary left shift ) acts on one-sided infinite sequence of binary numbers by


It just drops first digit of the sequence.


If we treat sequence as a binary fraction :


then shift map = the dyadic transformation = dyadic map = bit shift map= 2x  mod 1 map = Bernoulli map = doubling map = sawtooth map


and "shifting N places left is the same as multiplying by 2 to the power N (written as 2N)"[87] ( operator << )

In Haskell:

 shift k = genericTake q . genericDrop k . cycle  -- shift map

See also:


Multiplier of periodic z-point :[88][89]

  • "The value of is the same at any point in the orbit of a: it is called the multiplier of the cycle."[90]

Math notation :

Maxima CAS function for computing multiplier of periodic cycle :


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


It is used to :

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


complex number[edit]

  • numerical value : x+y*i
  • vector from origin to point ( x,y)
  • point (x,y) od 2D Cartesion plain

Rotation number[edit]

The rotation number[91][92][93][94][95] 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 [96]

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 number[edit]

  • of the map ( iterated function)[97][98]
    • "the winding number of the dynamic ray at angle a around the critical value, which is defined as follows: denoting the point on the dynamic a-ray at potential t greater or equal to zero by zt and decreasing t from +infinity to 0, the winding number is the total

change of arg(zt - c) (divided by 2*Pi so as to count in full turns).Provided that the critical value is not on the dynamic ray or at its landing point, the winding number is well-defined and finite and depends continuously on the parameter. " DIERK SCHLEICHER [99]


Orbit is a sequence of points[101]

  • phase space trajectories of dynamical systems
  • The orbit of periodic point is finite and it is called a cycle.



Forward orbit[102] of a critical point[103][104] is called a critical orbit. Critical orbits are very important because every attracting periodic orbit[105] attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.[106][107] [108]

This orbit falls into an attracting periodic cycle.

Code :

 This software is copyright (c) Conan Dalton 2008. Permission to use it is granted under the Perl Foundations's Artistic License 2.0.
 This software includes software that is copyright (c) Paul Graham and Robert Morris, distributed under the Perl Foundations's Artistic License 2.0.
 This software uses javacc which is copyright (c) its authors
(def plot (plt c)
  (with (z 0+0i
         n 0
         repeats 0)
    (while (and (small z) (< n 10000) (< repeats 1000))
      (assign n       (+ n 1)
              z       (+ c (* z z))
              repeats (if (apply plt (complex-parts z))
                          (+ repeats 1)

Here are images:

See also:


Homoclinic / heteroclinic[edit]


Inverse = Backward



  • set containing first n iterations of initial point without initial point and its k iterations
  • number of elements = n - k

It is used in the average colorings


  • set containing initial point and first n iterations of initial point
  • number of elements = n+1



  • point of parameter plane : " is renormalizable if restriction of some of its iterate gives a polinomial-like map of the same or lower degree. " [113]
  • parameter of the function


Period of point under the iterarted function f is the smallest positive integer value p for which this equality

holds is the period[114] of the orbit.[115]

is a point of periodic orbit ( limit cycle ) .

More is here


Planes [116]

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


  • the partition of the dynamic plane used in the definition of the kneading sequence: partition is formed by the dynamic rays at angles ϑ/2 and (ϑ + 1)/2, which land together at the critical point.
  • Markow


"A partition of a (finite) set is just a subdivision of the set into disjoint subsets. If the set is represented as points on a line (or around the edge of a disc), we can represent the partition with lines connecting the dots. The lines usually have lots of crossings. When the partition diagram has no crossing lines, it is called a non-crossing partition. ... They have a lot of beautiful algebraic structure, and are related to lots of old enumeration problems. More recently (and importantly), they turn out to be a crucial tool in understanding how the eigenvalues of large random matrices behave." Todd Kemp (UCSD)[117]

Key words:

  • Enumerative combinatorics


Dynamic plane[edit]

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

Parameter plane[edit]

See :[118]

Types of the parameter plane :

  • c-plane ( standard plane )
  • exponential plane ( map) [119][120]
  • flatten' the cardiod ( unroll ) [121][122] = "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)[123]
  • transformations [124]



the band-merging points are Misiurewicz points[125]


If there exist two distinct external rays landing at point we say that it is a biaccessible point.[126]


Nucleus or center of hyperbolic component[edit]

A center of a hyperbolic component H is a parameter ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." [127]

Synonyms :

  • Nucleus of a Mu-Atom [128]

How to find center/s ?

Center of Siegel Disc[edit]

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." [129]


A critical point[130] of is a point in the dynamical plane such that the derivative vanishes:



we see that the only (finite) critical point of is the point .

is an initial point for Mandelbrot set iteration.[131]


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).[132] 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.[133] 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.

The Feigenbaum Point[134] 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)[135]

Generalized Feigenbaum points are :

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

Examples :

  • -.1528+1.0397i)

The Mandelbrot set is conjectured to be self- similar around generalized Feigenbaum points[136] when the magnification increases by 4.6692 (the Feigenbaum Constant) and period is doubled each time[137]

n Period = 2^n Bifurcation parameter = cn Ratio
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[138]


Fibonacci point[139] [140][141]


The point at infinity [142]" 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[143]

Mandelbrot set at Fibonacci point


Misiurewicz point[144] = " parameters where the critical orbit is pre-periodic.


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

Parabolic point[edit]

parabolic points : this occurs when two singular points coallesce in a double singular point (parabolic point)[145]

"the characteristic parabolic point (i.e. the parabolic periodic point on the boundary of the critical value Fatou component) of fc"[146]


Point z has period p under f if :

In other words point is periodic

See also:


"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 [147]


A post-critical point is a point

where is a critical point.[148]


precritical points, i.e., the preimages of the critical point

reference point[edit]

Reference point of the image:

  • its orbit ( reference orbit) is computed with arbitrary precision and saved
  • orbits of the other points of the image ( no-reference points) are computed from reference orbit using standard precision ( with hardware floating point numbers ) = faster then using arbitrary precision

root or bond[edit]

The root point of the hyperbolic componnet of the Mandelbrot set :

  • A point where two mu-atoms meet
  • has a rotational number 0
  • it is a biaccesible point ( landing point of 2 external rays )



the singular points of a dynamical system

In complex analysis there are four classes of singularities:

  • Isolated singularities: Suppose the function f is not defined at a, although it does have values defined on U \ {a}.
    • The point a is a removable singularity of f if there exists a holomorphic function g defined on all of U such that f(z) = g(z) for all z in U \ {a}. The function g is a continuous replacement for the function f.
    • The point a is a pole or non-essential singularity of f if there exists a holomorphic function g defined on U with g(a) nonzero, and a natural number n such that f(z) = g(z) / (za)n for all z in U \ {a}. The least such number n is called the order of the pole. The derivative at a non-essential singularity itself has a non-essential singularity, with n increased by 1 (except if n is 0 so that the singularity is removable).
    • The point a is an essential singularity of f if it is neither a removable singularity nor a pole. The point a is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree.
  • Branch points are generally the result of a multi-valued function, such as or being defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The shape of the branch cut is a matter of choice, however, it must connect two different branch points (like and for ) which are fixed in place.


orbit portrait[edit]


There are two types of orbit portraits: primitive and satellite.[150] If is the valence of an orbit portrait and is the recurrent ray period, then these two types may be characterized as follows:

  • Primitive orbit portraits have and . Every ray in the portrait is mapped to itself by . Each is a pair of angles, each in a distinct orbit of the doubling map. In this case, is the base point of a baby Mandelbrot set in parameter space.
  • Satellite ( non-primitive ) orbit portraits have . In this case, all of the angles make up a single orbit under the doubling map. Additionally, is the base point of a parabolic bifurcation in parameter space.


Precision of :

  • data type used for computation. Measured in bits (width of significant ( fraction) = number of binary digits) or in decimal digits
  • input values
  • result ( number of significant figures )

See :

  • Numerical Precision : " Precision is the number of digits in a number. Scale is the number of digits to the right of the decimal point in a number. For example, the number 123.45 has a precision of 5 and a scale of 2."[151]
  • error [152]


small divisor problem[edit]


  • One-Dimensional Small Divisor Problems (On Holomorphic Germs and Circle Diffeomorphisms)

Where it can be found:

  • stability in mechanics, particularly in celestial mechanics
  • relations between the growth of the entries in the continued fraction expansion of t and the behaviour of f around z=0 under iteration.


Processes or transformations and phenomenona[edit]

Aliasing and antialising[edit]

Contraction and dilatation[edit]

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


Method of computing the derivative of a mathematical function


  • symbolic differentiation
  • Automatic Differentiation (AD)[153]
  • numeric differentiation [154][155][156] = the method of finite differences[157]


discretization[158] and its reverse [159]

Implosion and explosion[edit]

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)[160]

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 is the entire plane.[161]


  • changing from non-linear to linear
  • " ... turn the perturbated linear map into the exactly linear map ( it linearizes )" Jean-Christophe Yoccoz[162]


Mating [163]



  • " normalize this vector so it has modulus one " A Cheritat
  • move fixed point to the origin ( z= 0 )
  • mapping the range of variable to
    • [0.0, 1.0]
    • [2,255], like rgb values
  • converting closed curve to unit circle
  • converting closed curves to concentric circles with center at the origin[164]

See also:

  • uniformization
  • renormalization



"to any quadratic map f we can associate a canonical sequence of periods p1 < p2 <... for which f is renormalizable.

Depending on whether the sequence is:

  • empty
  • finite
  • infinite

the map f is called respectively:

  • non-renormalizable
  • at most finitely renormalizable
  • infinitely renormalizable" [168]


  • surgery in differential topology [169]





property or feature[edit]


density of the image[edit]

Dense image[171][172][173]

  • downsaling with gamma correction[174]
  • path finding[175]
  • supersampling: "ots of detail but fractal fades away as you get more accurate, as n increases in nxn supersampling" TGlad


sth is invariant with respect to the transformation = non modified, steady

Topological methods for the analysis of dynamical systems

Invariants type

  • metric invariants
  • dynamical invariants,
  • topological invariants.


Dynamical invariants = invariants of the dynamical system

Dynamical Invariants Derived from Recurrence Plots[176]


smooth = changing without visible (noticeable) edges


  • smooth gradient


  • conitnuous


  • discrete


  • stability of quasiperiodic motion under small perturbation. In the celestial mechanics dynamics of 3 bodies around sun is described by the system of differential equations. In such case it "becomes fantastically complicated and remains largely mysterious even today." See KAM = Kolmogorov–Arnold–Moser theorem and small divisor problem
  • stability of the fixed point under small perturbation
  • there is equivalence (for |f′(0)| ≤ 1) of stability (a topological notion) and linearizability (an analytical notion)


Radius of complex number[edit]

The absolute value or modulus or magnitude or radius of a complex number

Conformal radius[edit]

Conformal radius of Siegel Disk [177][178]

Escape radius ( ER)[edit]

Escape radius ( ER ) or bailout value is a radius of circle centered at origin ( z=0). This set is used as a target set in the bailout test ( escape time method = ETM )

Minimal Escape Radius should be grater or equal to 2 :

Better estimation is :[179][180]

Inner radius[edit]

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 radius[edit]

Internal radius is a:

  • absolute value of multiplier

See also : the N-2 rule[181]


A sequence is an ordered list of objects (or events).[182]

A series is the sum of the terms of a sequence of numbers.[183] Some times these names are not used as in above definitions.


is an itinerary of point x under the map f relative to the paritirtion.

It is a right-infinite sequence of zeros and ones [184]



Examples :

For rotation map and invariant interval ( circle ) :

one can compute  :


and split interval into 2 subintervals ( lower circle paritition):

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

Itinerary can be converted[185] to point

kneading sequence[edit]

  • "the kneading sequence of an external angle ϑ (here ϑ = 1/6) is defined as the itinerary of the orbit of ϑ under angle doubling, where the itinerary is taken with respect to the partition formed by the angles ϑ/2, and (ϑ + 1)/2 "[186]
  • The itinerary ν = ν1ν2ν3 . . . of the critical value is called the kneading sequence.[187] One can start from the critical point but neglect the initial symbol. Such sequence is computed with the Hubbard tree


Orbit can be:

  • forward = sequence of points
  • backward ( inverse )
    • tree in case of multivalued function
    • sequence


A series is the sum of the terms of a sequence of numbers.[188] Some times these names are not used as in above definitions.


  • Taylor series and Mandelbrot set[189]
  • The Existence and Uniqueness of the Taylor Series of Iterated Functions [190]





connected component (blob) in the image[edit]

Components of parameter plane[edit]


  • mu-atom[192]
  • ball
  • bud
  • bulb
  • decoration
  • lake
  • lakelet.[193]


Names :

  • mini Mandelbrot set
  • 'baby'-Mandelbrot set
  • island mu-molecules = embedded copy of the Mandelbrot Set[194]
  • Bug
  • Island
  • Mandelbrotie
  • Midget

List of islands :

Primitive and satellite[edit]

"Hyperbolic components come in two kinds, primitive and satellite, depending on the local properties of their roots." [195]

  • primitive ( non-satellite)
    • the root of component is not on the boundary of another component = "it was born from another hyperbolic component by the period increasing bifurcation"[196]
    • ones that have a cusp likes the main cardioid, when the little Julia sets are disjoint [197]
  • satellite
    • ones that don't have a cusp[198]
    • it's root is on the boundary of another hyperbolic component [199]
    • when the little Julia sets touch at their β-fixed



Child (Descendant ) and the parent ( ancestor)[edit]

  • ancestor of hyperbolic componnet
  • descendant of hyperbolic component = child [200]

Hyperbolic component of Mandelbrot set[edit]

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 has an attracting periodic orbit.

A center of a H is a parameter ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." [201]

A hyperbolic component is narrow if it contains no component of equal or lesser period in its wake [202]

features of hyperbolic component

  • period
  • islandhood ( shape = cardiod or circle )
  • angled internal address
  • lower and upper external angle of rays landing on it's root
  • center (
  • root
  • orientation
  • size


13/34 limb and wake on the left image

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

For every rational number , where p and q are relatively prime, a hyperbolic component of period q bifurcates from the main cardioid. The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called the p/q-limb. Computer experiments suggest that the diameter of the limb tends to zero like . The best current estimate known is the Yoccoz-inequality, which states that the size tends to zero like .

A period-q limb will have q − 1 "antennae" at the top of its limb. We can thus determine the period of a given bulb by counting these antennas.

In an attempt to demonstrate that the thickness of the p/q-limb is zero, David Boll carried out a computer experiment in 1991, where he computed the number of iterations required for the series to converge for z = ( being the location thereof). As the series doesn't converge for the exact value of z = , the number of iterations required increases with a small ε. It turns out that multiplying the value of ε with the number of iterations required yields an approximation of π that becomes better for smaller ε. For example, for ε = 0.0000001 the number of iterations is 31415928 and the product is 3.1415928.[203]


"what emerges from Myrrberg-Feigenbaum point is what we denominate a shrub due to its shape" M Romera


Wakes of Mandelbrot Set to Period 10

p/q-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 :

p/q-Subwake of W is a wake of a p/q-satellite component of W

Wake 1/3 bounded by 2 external rays and internal ray 1/3

wake is named after:

  • rotation number p/q ( as above)
  • angles of external rays landing in it's root point : "If two M-rays land at the same point we denote by wake the component of which does not contain 0."[204]

Components of dynamical plane[edit]

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 domain[edit]

"A Jordan domain[205] J is 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." [206]

Dwell bands[edit]

"Dwell bands are regions where the integer iteration count is constant, when the iteration count decreases (increases) by 1 then you have passed a dwell band going outwards (inwards). " [207] Other names:

  • level sets of integer escape time


Lea-Fatu flower


a partition of an interval into subintervals


sth is invariant if it does't change under transformation

"A subset S of the domain Ω is an invariant set for the system (7.1) if the orbit through a point of S remains in S for all t ∈ R. If the orbit remains in S for t > 0, then S will be said to be positively invariant. Related definitions of sets that are negatively invariant, or locally invariant, can easily be given" [209]

Examples :

  • invariant set
  • invariant point = fixed point
  • invariant cycle = periodic point
  • invariant circle
  • petal = invariant planar set

Level set[edit]

in case of :

level set is defined :

Boundaries of level sets ( lemniscates) are


Planar set[edit]

a non-separating planar set is a set whose complement in the plane is connected.[211]




Target set[edit]

How target set is changing along internal ray 0

Elliptic case[edit]

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 case[edit]

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 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 circle[edit]

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

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

Circle of radius=ER centered at the origin is :

Exterior of square[edit]

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

Parabolic case : petal[edit]

trap in parabolic case

In the parabolic case target set shoul be iside petal


Trap is 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 test[edit]

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.[213] It allows to find 2 sets :

  • escaping points ( it should be also the whole basing of attraction to infinity)[214]
  • 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 is in the target set then is escaping to infinity ( bailouts ) after n forward iterations ( steps).[215]

The output of test can be :

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

Types of bailout test:

Attraction test[edit]


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  101. wikipedia : Orbit_(dynamics)
  102. wikipedia : orbit (dynamics)
  103. Wikipedia : Complex quadratic polynomial - Critical point
  104. MandelOrbits - A visual real-time trace of Mandelbrot iterations by Ivan Freyman
  105. wikipedia : Periodic points of complex quadratic mappings
  106. M. Romera, G. Pastor, and F. Montoya : Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Fractalia 6, No. 21, 10-12 (1997)
  107. Burns A M : Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set. Mathematics Magazine, Vol. 75, No. 2 (Apr., 2002), pp. 104-116
  108. Khan Academy : Mandelbrot Spirals 2
  109. Complex Power Towers (Or ‘mucking around with Mathematica’) by Mike Croucher
  110. /DarkHeart by Chris King
  111. Alexandre Devert blog
  112. codeproject : Fractals-in-theory-and-practice
  113. Ouadratic-like maps and Renormalization by Nuria Fagella
  114. Peiod From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015.
  115. scholarpedia : Periodic Orbit for a Map
  116. wikipedia : Complex_quadratic_polynomial - Planes
  117. UC San Diego : MATH 196/296: Student Colloquium
  118. Alternate Parameter Planes by David E. Joyce
  119. mu-ency : exponential map by R Munafo
  120. Exponential mapping and OpenMP by Claude Heiland-Allen
  121. Linas Vepstas : Self Similar?
  122. the flattened cardioid of a Mandelbrot by Tom Rathborne
  123. Stretching cusps by Claude Heiland-Allen
  124. Twisted Mandelbrot Sets by Eric C. Hill
  125. doubling bifurcations on complex plane by E Demidov
  126. On biaccessible points in the Julia set of the family z(a+z^{d}) by Mitsuhiko Imada
  127. Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper
  128. Nucleus - From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015.
  129. Siegel disks by Xavier Buff and Arnaud Ch ́ritat e Univ. Toulouse Roma, April 2009
  130. wikipedia : Critical point (mathematics)
  131. Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations
  132. Cut point in wikipedia
  133. On local connectivity for the Julia set of rational maps : Newton’s famous example By P. Roesch
  134. muency : feigenbaum point
  135. On Periodic and Chaotic Regions in the Mandelbrot Set by G. Pastor , M. Romera, G. Álvarez, D. Arroyo and F. Montoya
  136. fractal-faq : section 6
  137. Period doubling and Feigenbaum's scaling be E Demidov
  138. mathoverflow question : is-there-a-way-to-find-regions-of-depth-in-the-mandelbrot-set-other-than-simply?rq=1
  139. Fractalforums : fibonacci-and-the-mandelbrot-set
  140. Parameter scaling for the Fibonacci point by Leroy Wenstrom
  141. The Fibonacci unimodal map by Mikhail Lyubich, John W. Milnor
  142. [w:Point at infinity|Point at infinity in wikipedia]
  143. Mathoverflow question : Attractive Basins and Loops in Julia Sets
  144. wikipedia : Misiurewicz point
  145. The bifurcation diagram of cubic polynomial vector fields on CP1 by Christiane Rousseau
  146. A rigidity result for some parabolic germs by Luna Lomonaco, Sabyasachi Mukherjee
  147. program Mandel by Wolf Jung , demo 2 page 3
  149. Bond the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018.
  150. wikipedia : Orbit portrait
  151. stackoverflow question : how-do-i-interpret-precision-and-scale-of-a-number-in-a-database
  152. taking-the-error-out-of-the-error-function by Fredrik Johansson
  154. wikipedia : Numerical_differentiation
  155. Numerical Recipies In C : 5.7 Numerical Derivatives
  156. stackoverflow question: is-there-any-standard-way-to-calculate-the-numerical-gradient
  157. wikipedia : Finite difference
  158. wikipedia : discretization
  159. mathinsight : from_discrete_to_continuous_dynamical_systems
  160. Airplane primitive parabolic implosion by Wolf Jung
  161. CANTOR BOUQUETS, EXPLOSIONS, AND KNASTER CONTINUA: DYNAMICS OF COMPLEX EXPONENTIALS by Robert L. Devaney Publicacions Matematiques, Vol 43 (1999), 27–54.
  162. Jean-Christophe Yoccoz Small Divisors: Number Theory in Dynamical Systems, page 46
  163. On The Notions of Mating by Carsten Lunde Petersen, Daniel Meyer
  164. towardsdatascience : gradient-descent-algorithm-and-its-variants
  165. Perturbation for the Mandelbrot set by DinkydauSet, Apr 28, 2014, 3:46:13 PM
  167. wikipedia: Approximation error
  168. Baby Mandelbrot sets, Renormalization and MLC Mikhail Lyubich
  169. wikipedioa: Surgery theory
  170. Tuning From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2014.
  171. wikipedi : dense_set
  172. mathoverflow question : is-there-an-almost-dense-set-of-quadratic-polynomials-which-is-not-in-the-inte/254533#254533
  173. fractalforums : dense-image
  174. A Cheritat wiki : see image showing gamma-correct downscale of dense part of Mandelbropt set
  175. fractal forums : pathfinding-in-the-mandelbrot-set/
  176. N. Marwan, M. C. Romano, M. Thiel, J. Kurths: Recurrence Plots for the Analysis of Complex Systems, Physics Reports, 438(5-6), 237-329, 2007.
  177. wikipedia : Conformal radius
  178. scholarpedia : Quadratic Siegel disks
  179. Julia Sets of Complex Polynomials and Their Implementation on the Computer by Christoph Martin Stroh
  180. fractalforums: bounding circle of julia sets by knighty
  181. The Mandelbrot Set and Julia Sets Combinatorics in the Mandelbrot Set - The 1/n2 rule, and deviations from it
  182. wikipedia : Sequence
  183. wikipedia : series
  184. Structure of Inverse Limit Spaces of Tent Maps with Nonrecurrent Critical Points by Brian Raines and Sonja Stimac
  185. Bifurcation structures in maps of Henon type by Kai T Hansen and Predrag Cvitanovic
  186. Hausdorff dimension of biaccessible angles of quadratic Julia sets and of the Mandelbrot set by Henk Bruin and Dierk Schleicher
  188. wikipedia : series
  189. quora: Why-is-the-Mandelbrot-set-a-fractal? Answer by Anders Kaseorg
  190. The Existence and Uniqueness of the Taylor Series of Iterated Functions by Daniel Geisler
  191. wikipedia : Continuum in set theory
  192. mu-atom From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018.  
  193. Mu-atom From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2013.
  194. Island Mu-Molecule by Robert P. Munafo, 2012 Aug 18.
  195. Internal addresses in the Mandelbrot set and Galois groups of polynomials by Dierk Schleicher, page 31
  196. notes by Lyubich
  197. Satellite copies of the Mandelbrot set by Luna Lomonaco
  198. mathoverflow : precise-location-of-the-mandelbrot-bulb-attachment-to-the-main-cardioid
  200. Child From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2013.
  201. Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper
  202. Internal addresses in the Mandelbrot set and irreducibility of polynomials by Dierk Schleicher
  203. Gary William Flake, The Computational Beauty of Nature, 1998. p. 125. ISBN 978-0-262-56127-3.
  204. Local properties of the Mandelbrot set at parabolic points by Tan Lei
  205. wikipedia : Carathéodory's theorem (conformal mapping)
  206. The intrinsic geometry of a Jordan domain by Richard L. Bishop
  207. fractalforums : binary-decomposition-and-external-angles by Claude
  208. math.stackexchange question : definition-of-markov-partition
  209. Norman Lebovitz : Textbook for Mathematics 27300
  210. wikipedia : Level set
  211. A. Blokh, X. Buff, A. Cheritat, L. Oversteegen The solar Julia sets of basic quadratic Cremer polynomials, Ergodic Theory and Dynamical Systems , 30 (2010), #1, 51-65
  212. Hyperoperations Wiki : Shell-Thron_region
  213. Fractus doc by Richard Rosenman
  214. wikipedia : Escaping set
  215. fractint doc : bailout