Fractals/Iterations in the complex plane/def cqp


Order is not only alphabetical but also by topic so use find ( Ctrl-f)

See also


 "Internal addresses encode kneading sequences in human-readable form, when extended to angled internal addresses they distinguish hyperbolic components in a concise and meaningful way. The algorithms are mostly based on Dierk Schleicher's paper Internal Addresses Of The Mandelbrot Set And Galois Groups Of Polynomials (version of February 5, 2008)" Claude Heiland-Allen[1]


  • finite / infinite
  • accesible/non-accesible
  • on the parameter plane / on th edynamic plane
  • simple/ angled
  • for Crossed Renormalizations[2]


Internal addresses describe the combinatorial structure of the Mandelbrot set.[3] It is one of the Analytical Naming Systems[4][5]


Internal address :

  • is not constant within hyperbolic component. Example : internal address of -1 is 1->2 and internal address of 0.9999 is 1[6]
  • of hyperbolic component is defined as a internal address of it's center


Angled internal address is an extension of internal address. The angled internal address of the end of a finite chain of child bulbs   would be:



  •   describes period 6 component which is a satelite of period 3 component.


  • islands
  • infinite sequence of bifurcations


Types of angleEdit

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
  • the boundary.

It is :

  • the same on all points on the external ray. It is important for proving connectedness of the Mandelbrot set.
  • a proper fraction
  • an approximation of directional derivative


The internal angle[7] is an angle of point of component's interior

  • it is a rational number and proper fraction measured in turns ( see multiplier map)
  • 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 ( inside child component)
  • "The internal angles start at 0, at the cusp, and increase counterclockwise. " Robert Munafo[8]



The plain angle is an agle of complex point = it's argument [9]


  • 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 :[10]


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[12]



  • 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.[14]
  • 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   is a boundary of unit disk[15]


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


Critical curvesEdit

Diagrams of critical polynomials are called critical curves.[16]

These curves create skeleton of bifurcation diagram.[17] (the dark lines[18])


  • a locally connected branched curve
  • "Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic."[19]
  • "a dendrite is a locally connected continuum that does not contain Jordan curves." [20]
  • "a locally connected continuum without subsets homeomorphic to a circle"
  • connected with no interior

See also :

  • Misiurewicz point on the parameter plane

Escape linesEdit

Escape line = boundary of escape time's level sets

"If the escape radius is equal to 2 the contour lines have a contact point (c= -2) and cannot be considered as equipotential lines" [21]



  • topological
  • shift invariants

examples :

"Quasi-invariant curves are used in the study of hedgehog dynamics" RICARDO PEREZ-MARCO[26]


  • field lines
    • external ray
    • internal ray


Equipotential linesEdit

Equipotential lines = Isocurves of complex potential

"If the escape radius is greater than 2 the contour lines are equipotential lines" [27]

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[28]


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[29][30]

It is a model of Mandelbrot or Julia set.

A lamination, L, is a union of leaves and the unit circle which satisfies :[31]

  • 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 [32]


Chords = leaves = arcs

A leaf on the unit disc is a path connecting two points on the unit circle.[33]

"In Thurston’s fundamental preprint, the two characteristic rays and their common landing point are the “minor leaf” of a “lamination”"[34]

Open curveEdit

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[35]

External rayEdit

The closure of an external ray is called a closed ray. If ray lands, then the closure of the ray is the union of the external ray and its landing point.[36]

Internal rayEdit


  • "The internal rays are the preimages of the radial segments under the coordinate with componenet center corresponding to 0." Alfredo Poirier[37]

Internal rays are :

  • dynamic ( on dynamic plane , inside filled Julia set )
  • |parameter ( on parameter plane , inside Mandelbrot set ) usuning multiplier map


For a parameter c with superattracting orbit: for every Fatou component   of filled julia set[38]   there is:

  • a unique periodic or pre-periodic point   of the super-attracting orbit
  • a Riemann map that maps:[39]

component to unit disc:


and point   to the origin:


The point   is called the center of component  .

For any angle   the pre-image of the radial segment of the unit disc


is called an internal ray of component   with well-defined landing point.


See also:


The internal rays are the curves that connects endpoints of external rays to the origin ( the only pole) by winding in the specific way through the Julia set. Unlike the external rays the internal rays allways cross other internal rays, usually at multiple points, hence they are interwined[40]


Escape routeEdit

Escape route is a path inside Mandelbrot set.

Escape route 1/2 <re>Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set by Anne M. Burns. Mathematics Magazine Vol. 75, No. 2 (Apr., 2002), pp. 104-116 </ref>

  • is part of the real slice of the mandelbrot set)
  • part of the real line x=0


  • start from center of period 1
  • go along internal ray 1/2 to root point of period 2 component
  • go along internal ray 0 to the center of period 2 component
  • go along internal ray 1/2 to root point of period 4 component
  • ...


A spider S is a collection of disjoint simple curves called legs [41]( extended rays = external + internal ray) in the complex plane connecting each of the post-critical points to infnity [42]

See :


In the case of complex_quadratic_polynomial   the spine   of the filled Julia set   is defined as arc between  -fixed point and  ,


with such properties:

  • spine lies inside  .[43] This makes sense when   is connected and full [44]
  • spine is invariant under 180 degree rotation,
  • spine is a finite topological tree,
  • Critical point   always belongs to the spine.[45]
  •  -fixed point is a landing point of external ray of angle zero  ,
  •   is landing point of external ray  .

Algorithms for constructing the spine:

  • detailed version is described by A. Douady[46]
  • Simplified version of algorithm:
    • connect   and   within   by an arc,
    • when   has empty interior then arc is unique,
    • otherwise take the shortest way that contains  .[47]

Curve   :


divides dynamical plane into two components.

Computing external angle for c from centers of hyperbolic components and Misiurewicz points:

 The spine of K is the arc from beta to minus beta. Mark 0 each time C is above the spine and 1 each time it is below. You obtain the expansion in base 2 of the external argument theta of z by C. This simply comes from the two following facts:  
 *  0 < theta < 1/2 if acces to z is above the spine,   1/2 < theta < 1 if it is below
 * function f doubles the external arguments with respect to K , as well as the potential, since  Riemman map ( Booettcher map) conjugates f to  .
 Note that if c and z are real, the tree reduces to the segment [beta',beta] of the real line, and the sequence of 0 and 1 obtained is just the kneading sequence studied by Milnor and Thurston (except for convention: they use 1 and -1). 
 This sequence appears now as the binary expansion of a number which has a geometrical interpretation. " A. Douady

Relation between spine and major leaf of the lamination


"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.


Distance Function

See also:

  • metric
  • Algorithm : Distance Estimation Method = DEM/J and DEM/M


  • symbolic[48][49][50]
  • complex [51][52]
  • Arithmetic
  • combinatorial
  • local/global
  • discrete/continous
  • parabolic/hyperbolic/eliptic


  • discrete local complex parabolic dynamics


"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



The Schwarzian DerivativeEdit

The Schwarzian Derivative [57][58][59][60]

Wirtinger derivativesEdit


the gradient is the generalization of the derivative for the multivariable functions[61][62]


  • (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[63]
  • (operator): Del or nabla is an gradient operator = a vector differential operator




See also

  • Gradient Descent Algorithm[64][65]
  • Gradient Ascent Algorithm
  • image gradient


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


Germ [66] 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 [68]
  • Iterated function = map[69]
  • an evolution function[70] of the discrete nonlinear dynamical system[71]

is called map  , examples:

  • rational maps
  • exponential maps
  • trigonometric maps
  • landing map : " A theorem of Caratheodory states that if   is a full compact and locally connected set, then external rays land and the landing map   is continuous."[72]

types or namesEdit


  • Brjuno function



An harmonic or spherical function is a:

  • "set of orthogonal functions all of whose curvatures are changing at the same rate."[73]
  • "harmonic functions relate two sets of different curves such that the rate of change of their respective curvatures is always equal. " and they are orthogonal
  • "One set of curves of the harmonic function expressed the pathways of minimal change in the potential for action, while the other, orthogonal curves expressed the pathways of maximum change in the potential for action."
  • "a pair of harmonic conjugate functions, u and v. They satisfy the Cauchy-Riemann equations. Geometrically, this implies that the contour lines of u and v intersect at right angles"[74]

Geometric examples:

  • " A set of concentric circles and radial lines comprises an harmonic function because both the circles and the radial lines intersect orthogonally and both have constant curvature."
  • "a set of orthogonal ellipses and hyperbolas."

How to find harmonic conjugate function ? [75]


meromorphic maps: Those with NO FINITE, NON-ATTRACTING FIXED POINTS[76]



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 finiteEdit

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[77][78][79]

Yoccoz’s functionEdit


Interior of the Cauliflower Julia set. .The black structure around fixed point and it's preimages is a numerical error ( glitch) }}


  • Incorrect parts of renders[80] using perturbation techique
  • pixels which dynamics differ significantly from the dynamics of the reference pixel[81]"These can be detected and corrected by using a more appropriate reference."[82]



Dessin d'enfantEdit

See also:


Farey treeEdit

Farey tree = Farey sequence as a tree

Hubbard treeEdit

  • a simplified, combinatorial model of the Julia set ( MARY WILKERSON)
  • "Hubbard trees are finite planar trees, equipped with self-maps, which classify postcritically finite polynomials as holomorphic dynamical systems on the complex plane." [83]
  • " 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[84]

Rooted treeEdit

rooted tree of preimages:


where a vertex   is connected by an edge with  .




  • magnitude of the point ( complex number in 2D case) = it's distance from the origin[85]
  • 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.[86]

Complex quadratic mapEdit


c form :  Edit

quadratic map[87]

  • 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 formsEdit

Start from :

  • internal angle  
  • internal radius r

Multiplier of fixed point :


When one wants change from lambda to c :[88]


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[89]

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 [90]

  • irrational rotation[91]

Doubling mapEdit

definition [92][93][94]

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[95]
-- 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


is half a turn = 180 degrees = Pi radians.

Images and preimages under doubling map d

Feigenbaum mapEdit

"the Feigenbaum map F is a solution of Cvitanovic-Feigenbaum equation"[97]

First return mapEdit

"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 "[99]

"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." [100]

"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"[101]

postcritically finiteEdit

postcritically finite: maps whose critical orbits are all periodic or preperiodic[102]

  " In the theory of iterated rational maps, the easiest maps to understand are postcritically finite: maps whose critical orbits are all periodic or preperiodic. These maps are also the most important maps for understanding the combinatorial structure of parameter spaces of rational maps. "

A postcritically finite quadratic polynomial fc(z) = z^2+c may be:[103]

  • periodic of satellite type
  • periodic of primitive type
  • critically preperiodic (Misiurewicz type)

Examples are given by:

  • the Basilica Q(z) = z^2 − 1
  • the Kokopelli (
  • P(z) = z^2 + i ( dendrite)

Critically preperiodic polynomialsEdit

  • the critical point of fc is strictly preperiodic
  • parameter c is from Thurston-Misiurewicz points–values on the boundary of the Mandelbrot set = Misiurewicz point
  • Julia set is dendrite

Multiplier mapEdit

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.[104]

It can be computed using :


Riemann mapEdit

Riemann mapping theorem[105] 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[106]
      • for the Fatou component containing a superattracting fixed point for a rational map[107]
      • 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)[108]
    • 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[109]

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 mapEdit

     "If a is rational, then every point is periodic. If a is irrational, then every point has a dense orbit." David Richeson[110]


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


It is used for computing :


Shift mapEdit

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)"[111] ( operator << )

In Haskell:

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

See also:


Multiplier of periodic z-point :[112][113]

  • "The value of   is the same at any point in the orbit of a: it is called the multiplier of the cycle."[114]

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

" The multiplier of a fixed point gives information about its stability (the behaviour of nearby orbits )" [115]


complex numberEdit

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


Fegenbaum constantEdit

  • first ( delta)[116]
  • second ( alpha)

How to compute:


It hase many meanings:[117]

  • unit of the angle
  • degree of a function
    • polynomial
    • rational function[118]

Rotation numberEdit

The rotation number[119][120][121][122][123] 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 [124]

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

  • of the map ( iterated function)[125][126]
    • "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 [127]
    • "the winding number of the dynamic ray at angle ϑ around the critical value, which is defined as follows: denoting the point on the dynamic ϑ-ray at potential t ≥ 0 by zt and decreasing t from +∞ to 0, the winding number is the total change of arg(zt − c) (divided by 2π 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. When the parameter c moves in a small circle around c0 and if the winding number is defined all the time, then it must change by an integer corresponding to the multiplicity of c as a root of z(c) − c. However, when the parameter returns back to where it started, the winding number must be restored to what it was before. This requires a discontinuity of the winding number, so there are parameters arbitrarily close to c0 for which the critical value is on the dynamic ray at angle ϑ, and c0 is a limit point of the parameter ray at angle ϑ. Since this parameter ray lands, it lands at c0."
  • of the curve [128][129]
    • the winding number of a curve is the number of complete rotations, in the counterclockwise sense, of the curve around the point(0, 0).[130]
    • w(γ, x) = number of times curve γ winds round point x. The winding number is signed: + for counterclockwise, − for clockwise.[131]

Computing winding number of the curve ( which is not crossing the origin ) using:

  • numerical integration
  • computational geometry

The discrete winding number = winding number of polygon aproximating curve


Orbit is a sequence of points[132]

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



Forward orbit[133] of a critical point[134][135] is called a critical orbit. Critical orbits are very important because every attracting periodic orbit[136] attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.[137][138][139]






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 / heteroclinicEdit


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





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


holds is the period[145] of the orbit.[146]

  is a point of periodic orbit ( limit cycle )  .

More is here


Planes [147]

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


In topology: two-dimensional sphere = 2-sphere = the two-dimensional surface of a three-dimensional ball[148]

Geometrically, the set of extended complex numbers is referred to as the Riemann sphere or extended complex plane.



  • Markow
  • Yoccoz puzzle

Kneading paritition of the dynamic planeEdit

In case of critically preperiodic polynomials the partition of the dynamic plane used in the definition of the kneading sequence.

Partition is formed by the dynamic rays at angles:

  • t/2
  • (t + 1)/2

which land together at the critical point.

Angle t is angle which lands on the critical value:


Spine paritition of the dynamic planeEdit

Curve   :



divides dynamical plane into two components.


noncrossing: "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)[149]

Key words:

  • Enumerative combinatorics


  • slit plane = plane with the slit deleted[150] : Let S be the "slit plane"  
  • chessboard or checkerboards

types in case of discrete dynamical systemEdit

Dynamic plane or phase spaceEdit

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

Parameter planeEdit

See :[151]

Types of the parameter plane :

  • c-plane ( standard plane )
  • exponential plane ( map) [152][153]
  • flatten' the cardiod ( unroll ) [154][155] = "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)[156]
  • transformations [157]



the band-merging points are Misiurewicz points[158]


If there exist two distinct external rays landing at point we say that it is a biaccessible point.[159]

blowup pointEdit

blowup point = parameter for which the critical orbits map to ∞, so the Julia set is the entire sphere [160]


" a point of the Julia set is buried if it is not in the boundary of any Fatou component." [161]

polynomials do not have buried points

some rational Julia sets have ( Residual Julia Set = Buried Points )


Nucleus or center of hyperbolic componentEdit

A center of a hyperbolic component H is a parameter   ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." [162]

Synonyms :

  • Nucleus of a Mu-Atom [163]

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." [164]


A critical point[165] of   is a point   in the dynamical plane such that the derivative vanishes:


A critical value is an image of critical point

complex quadratic polynomialEdit

For the complex quadratic polynomial in the c form




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

  is an initial point for Mandelbrot set iteration.[166]


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).[167] 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.[168] Cut rays can be used to construct puzzles.

Cut angle is an angle of cut ray.


Periodic point when period = 1


The Feigenbaum Point[169] is a :

  • point c of parameter plane
  • is the limit of the period doubling cascade of bifurcations
  • the accumulation point of the period-doubling cascade in the real-valued x^2+c mapping
  • an infinitely renormalizable parameter of bounded type
  • boundary point between chaotic ( -2 < c < MF ) and periodic region ( MF< c < 1/4)[170]


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[171] when the magnification increases by 4.6692 (the Feigenbaum Constant) and period is doubled each time[172]

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[173]


Fibonacci point[174] [175][176]


The point at infinity [177]" 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[178]

Mandelbrot set at Fibonacci point


Misiurewicz point[179] = " parameters where the critical orbit is pre-periodic.


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)[180]

"the characteristic parabolic point (i.e. the parabolic periodic point on the boundary of the critical value Fatou component) of fc"[181]


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 )

other names

  • pinch points
  • cut points

See for examples :

  • period 2 = Mandel, demo 2 page 3.
  • period 3 = Mandel, demo 2 page 5 [182]


"A point in the dendrite is called a pool if it is the landing point for two external rays, both of whose angles are of the form


for some k, n ∈ N, where k ≡ 1 mod 6.


central pool ... it is geometrically the center of the dendrite; a one half rotation around this point maps the dendrite to itself." [183]


A post-critical point is a point


where   is a critical point.[184]

See also :


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

reference pointEdit

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


point of the parameter plane " is renormalizable if restriction of some of its iterate gives a polinomial-like map of the same or lower degree. " [185]

infinitely renormalizableEdit

" 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[186]


Virtually repellingEdit

virtually repelling fixed points[187]

root or bondEdit

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.


  • from Mu-Ency : "the point in a primary filament that has the simplest external angle; this is the point that you get by appending FS[(1/2B1)] an infinite number of times to the primary filament's name." This is also the "limit" of the ... series.
  • Misurewicz point


"A point in the dendrite is called a triple point if its removal separates the dendrite into three connected components. Such a point is the landing point for three external rays, whose angles all have of the form


for some k, n ∈ N, where k is congruent to 1, 2 or 4, mod 7." Will Smith in Thompson-Like Groups for Dendrite Julia Sets


orbit portraitEdit


There are two types of orbit portraits: primitive and satellite.[189] 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.


Critical orbit portrait = portrait of the critical orbit

... for the polynomial   we may note the critical orbit portrait:


for this map, or we may double the angles of external rays and record the locations of landing points in order to observe the same behavior." [190]


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."[191]
  • error [192]


Douady’s principleEdit

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


small divisor problemEdit


  • One-Dimensional Small Divisor Problems[193] (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 phenomenonaEdit

Aliasing and antialisingEdit

Contraction and dilatationEdit

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


Method of computing the derivative of a mathematical function


  • symbolic differentiation
  • Automatic Differentiation (AD)[195]
  • numeric differentiation [196][197][198] = the method of finite differences[199]


  • discretization[200] and its reverse [201]
  • discretize/homogenize in the DDG ( Discrete Differential Geometry)

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)[202]
    • " We can see that   looks somewhat like   from the "outside", but on the "inside" there are curlicues; pairs of them are vaguely reminiscent of "butterflies". As t→0, these butterflies persist and remain uniformly large. We think of t as representing time, which decreases to 0. The fact that they suddenly disappear for t=0 is the phenomenon called "implosion". Or, if we think of time starting at t=0, then the instantaneous appearance of large "butterflies" for t>0 may be thought of as "explosion". "
  • Semi-parabolic implosion in  [203]

Explosion 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.[204]


  • integrating along some vector field means finding a solution curve. Example : finding extrrernal ray using Runge-Kutta method for numerical integration[205]


  • changing from non-linear to linear
  • " ... turn the perturbated linear map   into the exactly linear map   ( it linearizes   )" Jean-Christophe Yoccoz[206]
  • linearization in english wikipedia
  • Linearization in scholarpedia
  • "System is linearizable at the origin if and only if there exists a change of coordinates which linearizes the system, that is, all the coefficients of the normal form vanish." [207]
Linearization with inverse function


  • Parabolic Linearization


Mating [208]



  • normalize = transformation to the model[209]
  • " normalize this vector so it has modulus one " A Cheritat
  • move fixed point to the origin ( z= 0 )
  • mapping the range of variable to standard range
    • [0.0, 1.0]
    • [0,255], like rgb values
  • converting closed curve to unit circle
  • converting closed curves to concentric circles with center at the origin[210]

See also:

  • uniformization
  • renormalization


  • Parametrization is the process of finding parametric equations of a curve[211]



"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" [215]

"Sectorial renormalizations are useful in the nonlinearizable situation. " Ricardo Pérez-Marco[216]


  • surgery in differential topology [217]
  • regluing [218]




Uniformization of 
  • Hyperbolic Components of Mandelbrot set to the unit disc = multiplier map
  • basin of superattractive fixed point - Bottcher map (The Bottcher uniformization theorem)


property or featureEdit


density of the imageEdit

Dense image[220][221][222]

  • downsaling with gamma correction[223]
  • path finding[224]
  • supersampling: "ots of detail but fractal fades away as you get more accurate, as n increases in nxn supersampling" TGlad


The meaning of the terms "elliptic, hyperbolic, parabolic" in different disciplines in mathematics[225]


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[226]


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)

Compare with:


Radius of complex numberEdit

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

Conformal radiusEdit

Conformal radius of Siegel Disk [227][228]

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 :[229][230]



How to choose parameters for which Level curves cross critical point ( and it's preimages ) ? Choose escape radius equal to n=th iteration of critical value.

// find such ER for LSM/J that level curves croses critical point and it's preimages
double GiveER(int i_Max){

	complex double z= 0.0; // criical point
	int i;
	 ; // critical point escapes very fast here. Higher valus gives infinity
	for (i=0; i< i_Max; ++i ){
		z=z*z +c; 
	 return cabs(z);

Another way: choose the parameter c such that it is on an escape line, then the critical value will be on an escape line as well.

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:

  • absolute value of multiplier  

See also : the N-2 rule[231]


A sequence is an ordered list of objects (or events).[232]

A series is the sum of the terms of a sequence of numbers.[233] 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 [234]



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[235] to point  


kneading sequenceEdit

  • "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 "[236]
  • The itinerary ν = ν1ν2ν3 . . . of the critical value is called the kneading sequence.[237] One can start from the critical point but neglect the initial symbol. Such sequence is computed with the Hubbard tree

See also:

Thue–Morse sequenceEdit

Thue–Morse sequence


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.[239] Some times these names are not used as in above definitions.


  • Taylor series and Mandelbrot set[240]
  • The Existence and Uniqueness of the Taylor Series of Iterated Functions [241]


Attracting setEdit

Informal definition :

  "an attracting set for a dynamical system is a closed subset A of its phase space such that for "many" choices of initial point the system will evolve towards A ." John W Milnor[242]




chaotic bandEdit

period-  chaotic band  [244]

  • is between Misiurewicz points (primary separators)   and  
  • it's biggest midget has period  
  • contains Sharkovsky subsequence : sequence of islands for periods :   for k = 1, 2, ..... (in the increasing order = increasing from left to right). These are first appearence of hyperbolic components with such period in Sharkowsky ordering
  • is on n-place in Sharkowsky ordering




primary separators ( Misiurewicz points )

  •   = tip of main antenna , external angle = 1/2[245]
  •  , external angles 5 and 7/12



Basin can consist of

  • one component, like basin of infinity

of attractionEdit


  • An attractor's basin of attraction is the region of the phase space, over which iterations are defined, such that any point (any initial condition) in that region will asymptotically be iterated into the attractor
  • The collection of all points whose iterates under f converge to the attractor [246]

immediate basin of attractionEdit

the component of the basin containing the periodic point itself


  • basin of infinity ( whole basin = one component)



connected component (blob) in the imageEdit

Components of parameter planeEdit


  • mu-atom[247]
  • ball
  • bud
  • bulb
  • decoration : "A decoration of the Mandelbrot set M is a part of M cut off by two external rays landing at some tip of a satellite copy of M attached to the main cardioid."[248]
  • lake
  • lakelet.[249]


from Mu-Ency: "Any contiguous subset of the Mandelbrot Set which consists of the infinitely convoluted and branching structures that connect the island mu-molecules to each other."

Some colloquial names for filaments:

  • antenna
    • main antenna
  • spike
  • spoke.
 "A filament consists of a) minibrots and b) limit points of sequences of those minibrots. The latter include Misiurewicz points (rational external angles, one for filament termini and two or more for interior points such as multi-armed spiral centers) and other points (with irrational external angles).
 My intuition says if you zoom to a succession of smaller minis along a filament, if this is done in a pattern for infinitely long you tend to a Misiurewicz point, and if it's done randomly for infinitely long you tend to an irrational point. But I have no proof of this. 
 Other noninterior points on filaments mostly belong to individual minibrots: cardioid cusps (two rational external angles, odd denominator) and minibrot-filament branch tips (Misiurewicz points, two rational external angles, even denominator). 
 There is one last point: the exact base of the filament where it attaches to something (minibrot or main set). This point has irrational external angles. The Feigenbaum point at the base of the spike is one of these." pauldelbrot[250]


Names :

  • mini Mandelbrot set
  • 'baby'-Mandelbrot set
  • island mu-molecules = embedded copy of the Mandelbrot Set[251]
  • Bug
  • Island
  • Mandelbrotie
  • Midget

List of islands :

Primitive and satelliteEdit

"Hyperbolic components come in two kinds, primitive and satellite, depending on the local properties of their roots." [252]

  • 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"[253]
    • ones that have a cusp likes the main cardioid, when the little Julia sets are disjoint [254]
  • satellite
    • ones that don't have a cusp[255]
    • it's root is on the boundary of another hyperbolic component [256]
    • when the little Julia sets touch at their β-fixed point


Child (Descendant ) and the parent ( ancestor)Edit

  • ancestor of hyperbolic componnet
  • descendant of hyperbolic component = child [257]

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   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." [258]

A hyperbolic component is narrow if it contains no component of equal or lesser period in its wake [259]

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


  • LAHCs = the last appearance HCs placed in the chaotic region


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.[260]


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


"Colloquial term for a filament, specifically one of the "arms" radiating from a branch point." - from Mu-Ency


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."[261]

Components of dynamical planeEdit

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

For a quadratic polynomial with a parabolic orbit, the unique Fatou component[262] containing the critical value will be called the characteristic Fatou component; (Dierk Schleicher in Rational Parameter Rays of the Mandelbrot Set )


Domain in mathematical analysis it is an open connected set

Jordan domainEdit

"A Jordan domain[263] 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." [264]

Dwell bandsEdit

"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). " [265] 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" [267]

Examples :

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

Level setEdit

in case of :

level set is defined :


Boundaries of level sets ( lemniscates) are




The Cantor locus is the unique hyperbolic component, in the moduli space of quadratic rational maps rat2, consisting of maps with totally disconnected Julia sets [269]


In one-dimensional complex dynamics, the connectedness locus is a subset of the parameter space of rational functions, which consists of those parameters for which the corresponding Julia set is connected. the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps.

Planar setEdit

a non-separating planar set is a set whose complement in the plane is connected.[270]


"The postsingular set P(f) of a meromorphic function f is the closure of the union of forward iterates of the singular set S(f):"[271]



  • the iterates of the critical set
  • "For a rational map of the Riemann sphere f, the post-critical set PC(f) is defined as closure of orbits of all critical points of f. It is proved by Lyubich [Ly83b] that the post-critical set of a rational map is the measure theoretic attractor of points in the Julia set of that map. That is, for every neighborhood of the post-critical set, orbit of almost every point in the Julia set eventually stays in that neighborhood" [272]
  • "The postcritical set P(f) of a rational map f is the smallest forward invariant subset of that contains the critical values of f."[273]
  • "The analysis of the post-critical set plays a central role in the dynamics of rational maps, mainly because of the following two properties:
    • the set of attracting cycles is always finite for rational maps f
    • every attracting cycle attracts the orbit of a critical point of f."[274]




Singular setEdit

"The singular set S(f) of a meromorphic function f : C → Cˆ is the collection of values w at which one can not define all branches of the inverse f −1 in any neighborhood of w. If f is rational, then S(f) coincides with the collection of critical values of f. If f is transcendental meromorphic, f −1 may also fail to be defined in a neighborhood of an asymptotic value" [276]

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   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   and radius =ER :


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

Circle of radius=ER centered at the origin is :  

Exterior of squareEdit

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


Parabolic case : petalEdit

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 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.[277] It allows to find 2 sets :

  • escaping points ( it should be also the whole basing of attraction to infinity)[278]
  • 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).[279]

The output of test can be :

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

Types of bailout test:

Attraction testEdit


  • The Douady-Hubbard landing theorem for periodic external rays of polynomial dynamics: "for a complex polynomial f with bounded postcritical set, every periodic external ray lands at a repelling or parabolic periodic point, and conversely every repelling or parabolic point is the landing point of at least one periodic external ray." [280]


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  235. Bifurcation structures in maps of Henon type by Kai T Hansen and Predrag Cvitanovic
  236. Hausdorff dimension of biaccessible angles of quadratic Julia sets and of the Mandelbrot set by Henk Bruin and Dierk Schleicher
  238. rosetta code : Thue-Morse
  239. wikipedia : series
  240. quora: Why-is-the-Mandelbrot-set-a-fractal? Answer by Anders Kaseorg
  241. The Existence and Uniqueness of the Taylor Series of Iterated Functions by Daniel Geisler
  242. : Attractor
  243. wikipedia : Continuum in set theory
  244. Harmonic structure of one-dimensional quadratic maps by G. Pastor, M. Romera, and F. Montoya
  245. G. Pastor, M. Romera, G. Álvarez and F. Montoya, "Operating with external arguments in the Mandelbrot set antenna", Physica D, 171 (2002), 52-71
  246. Alicia Cordero, Fazlollah Soleymani, Juan R. Torregrosa, Stanford Shateyi, "Basins of Attraction for Various Steffensen-Type Methods", Journal of Applied Mathematics, vol. 2014, Article ID 539707, 17 pages, 2014.
  247. mu-atom From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018.  
  248. A priori bounds for some infinitely renormalizable quadratics: II. Decorations by Jeremy Kahn, Mikhail Lyubich
  249. Mu-atom From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2013.
  250. : how-thick-are-the-filaments
  251. Island Mu-Molecule by Robert P. Munafo, 2012 Aug 18.
  252. Internal addresses in the Mandelbrot set and Galois groups of polynomials by Dierk Schleicher, page 31
  253. notes by Lyubich
  254. Satellite copies of the Mandelbrot set by Luna Lomonaco
  255. mathoverflow : precise-location-of-the-mandelbrot-bulb-attachment-to-the-main-cardioid
  257. Child From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2013.
  258. Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper
  259. Internal addresses in the Mandelbrot set and irreducibility of polynomials by Dierk Schleicher
  260. Gary William Flake, The Computational Beauty of Nature, 1998. p. 125. ISBN 978-0-262-56127-3.
  261. Local properties of the Mandelbrot set at parabolic points by Tan Lei
  262. wikipedia: Classification_of_Fatou_components
  263. wikipedia : Carathéodory's theorem (conformal mapping)
  264. The intrinsic geometry of a Jordan domain by Richard L. Bishop
  265. fractalforums : binary-decomposition-and-external-angles by Claude
  266. math.stackexchange question : definition-of-markov-partition
  267. Norman Lebovitz : Textbook for Mathematics 27300
  268. wikipedia : Level set
  269. Limits of Polynomial-like Quadratic Rational Maps III: The Cantor Locus by Eva Uhre
  270. 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
  271. Prescribing the Postsingular Dynamics of Meromorphic Functions by Christopher J. Bishop, Kirill Lazebnik
  272. Dynamics of complex unicritical polynomials. A Dissertation Presented by Davoud Cheraghi
  273. On the postcritical set of a rational map by Laura G. DeMarco, Sarah C. Koch and Curtis T. McMullen
  274. Dynamics of hyperbolic correspondences by Carlos Siqueira
  275. Hyperoperations Wiki : Shell-Thron_region
  276. Prescribing the Postsingular Dynamics of Meromorphic Functions by Christopher J. Bishop, Kirill Lazebnik
  277. Fractus doc by Richard Rosenman
  278. wikipedia : Escaping set
  279. fractint doc : bailout
  280. A landing theorem for entire functions with bounded post-singular sets by Anna Miriam Benini, Lasse Rempe-Gillen