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Fractals/Iterations in the complex plane/def cqp

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Definitions

Contents

AddressEdit

InternalEdit

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

angledEdit

Angled internal address is an extension of internal address

  

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

AngleEdit

Types of angleEdit

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

where :

externalEdit

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

internalEdit

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]

 

plainEdit

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

UnitsEdit

  • 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 :[6]

 

where :

 

BifurcationEdit

  • Numerical Bifurcation Analysis of Maps

CoordinateEdit

Coordinate :

CurvesEdit

Types:

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

Description[8]

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

closedEdit

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.[9]
  • 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"

CircleEdit

Inner circleEdit

Unit circleEdit

Unit circle   is a boundary of unit disk[10]

 

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

 

Critical curvesEdit

Diagrams of critical polynomials are called critical curves.[11]

These curves create skeleton of bifurcation diagram.[12] (the dark lines[13])

Escape linesEdit

"If the escape radius is equal to 2 the contour lines have a contact point (c= -2) and cannot be considered as equipotential lines" [14]

InvariantEdit

Types:

  • topological
  • shift invariants

examples :

  • curve is invariant for the map f ( evolution function ) if images of every point from the curve stay on that curve[15][16][17]
  • curve is invariant for a system of ordinary differential equations[18]

IsocurvesEdit

Equipotential linesEdit

Equipotential lines = Isocurves of complex potential

"If the escape radius is greater than 2 the contour lines are equipotential lines" [19]

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[20]

LaminationEdit

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[21][22]

It is a model of Mandelbrot or Julia set.

A lamination, L, is a union of leaves and the unit circle which satisfies :[23]

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

LeafEdit

Chords = leaves = arcs

A leaf on the unit disc is a path connecting two points on the unit circle.[24]

Open curveEdit

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

RayEdit

Rays are :

  • invariant curves
  • dynamic or parameter
  • external or internal

External rayEdit

Internal rayEdit

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

Internal rays are :

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

SpiderEdit

A spider S is a collection of disjoint simple curves called legs [25]( extended rays = external + internal ray) in the complex plane connecting each of the post-critical points to infnity [26]

See :

VeinEdit

"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 )

DensityEdit

density of the imageEdit

Dense image[27][28][29]

  • downsaling with gamma correction[30]
  • path finding[31]
  • supersampling: "ots of detail but fractal fades away as you get more accurate, as n increases in nxn supersampling" TGlad

DerivativeEdit

DiscretizationEdit

discretization[33] and its reverse [34]

DiscriminantEdit

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.

DynamicsEdit

symbolicEdit

"Symbolic dynamics encodes :

equationEdit

differentialEdit

differential equations

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

FunctionEdit

DerivativeEdit

Derivative of Iterated function (map)[39]

Derivative with respect to cEdit

On parameter plane :

  •   is a variable
  •   is constant
 

This derivative can be found by iteration starting with

 
 

and then

 
 

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 zEdit

  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[40]

The Schwarzian DerivativeEdit

The Schwarzian Derivative [41]

GermEdit

Germ [42] of the function f in the neighborhood of point z is a set of the functions g which are indistinguishable in that neighborhood

 

See :

mapEdit

  • differences between map and the function [43]
  • Iterated function = map[44]
  • an evolution function[45] of the discrete nonlinear dynamical system[46]
 

is called map   :

 

typesEdit

PolynomialEdit

CriticalEdit

Critical polynomial :

 

so

 

 

 

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

meromorphicEdit

meromorphic maps: Those with NO FINITE, NON-ATTRACTING FIXED POINTS[47]

ResurgentEdit

"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[48]

glitchesEdit

glitches = Incorrect parts of renders[49] using perturbation techique

grafEdit

Dessin d'enfantEdit

TreeEdit

Farey treeEdit

Farey tree = Farey sequence as a tree

Hubbard treeEdit

"Hubbard trees are finite planar trees, equipped with self-maps, which classify postcritically finite polynomials as holomorphic dynamical systems on the complex plane." [50]

InvariantsEdit

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.

dynamicalEdit

Dynamical invariants = invariants of the dynamical system

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

Dynamical Invariants Derived from Recurrence Plots[51]

IntervalEdit

a partition of an interval into subintervals

  • Markov paritition[52]

IterationEdit

ItineraryEdit

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

It is a right-infinite sequence of zeros and ones [53]

  

where

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[54] to point  

 

MagnitudeEdit

  • magnitude of the point ( complex number in 2D case) = it's distance from the origin[55]
  • radius is the absolute value of complex number ( compare to arguments or angle)

MapEdit

description

typesEdit

  • The map f is hyperbolic if every critical orbit converges to a periodic orbit.[56]

Complex quadratic mapEdit

FormsEdit

c form :  Edit

quadratic map[57]

  • math notation :  
  • Maxima CAS function :
f(z,c):=z*z+c;
(%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 :[58]

 

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[59]

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 */
F(w):=w/2-w*w/4;

/* 
circle D={w:abs(w)=1 } where w=l(t,r) 
t is angle in turns ; 1 turn = 360 degree = 2*Pi radians 
r is a radius 
*/
ToCircle(t,r):=r*%e^(%i*t*2*%pi);

GiveC(angle,radius):=
(
 [w],
 /* point of  unit circle   w:l(internalAngle,internalRadius); */
 w:ToCircle(angle,radius),  /* point of circle */
 float(rectform(F(w)))    /* point on boundary of period 1 component of Mandelbrot set */
)$

compile(all)$

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

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

Circle mapEdit

Circle map [60]

  • irrational rotation[61]

Doubling mapEdit

definition [62]

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

DoublingMap(r):=
  block([d,n],
        n:ratnumer(r),
        d:ratdenom(r),
        mod(2*n,d)/d)$
  • 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[63]
-- 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. 
//   http://mndynamics.com/indexp.html 
// 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"[65]

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 "[67]

"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." [68]

"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"[69]

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:

 

where:

  • 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.[70]

It can be computed using :

Riemann mapEdit

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

 

where:

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

Examples

  • 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[71]
      • for the Fatou component containing a superattracting fixed point for a rational map[72]
      • 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

function:

  • explicit formula ( only in simple cases)
  • numerical aproximation ( in most of the cases)[73]
    • 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[74]

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

  

Then

  •  
  •  
  • and   is harmonic

so:

  • 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

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)"[75] ( operator << )

In Haskell:

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

See also:

MultiplierEdit

Multiplier of periodic z-point :[76][77]

  • "The value of   is the same at any point in the orbit of a: it is called the multiplier of the cycle."[78]

Math notation :

 

Maxima CAS function for computing multiplier of periodic cycle :

m(p):=diff(fn(p,z,c),z,1);

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

period    
1    
2    
3    



It is used to :

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

NormalizeEdit

  • " normalize this vector so it has modulus one " A Cheritat

NumberEdit

Rotation numberEdit

The rotation number[79][80][81][82][83] 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 [84]

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

def[85]

OrbitEdit

Orbit is a sequence of points[86]

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

BackwardEdit

CriticalEdit

Forward orbit[87] of a critical point[88][89] is called a critical orbit. Critical orbits are very important because every attracting periodic orbit[90] attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.[91][92] [93]

 

 

 

 

 

This orbit falls into an attracting periodic cycle.

Code :

"https://github.com/conanite/rainbow/blob/master/src/arc/rainbow/spiral.arc
 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)
                          0)))))

Here are images:

ForwardEdit

Homoclinic / heteroclinicEdit


InverseEdit

Inverse = Backward


periodicEdit

skippedEdit

  • 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

truncatedEdit

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

 

ParameterEdit

Parameter

  • point of parameter plane : " is renormalizable if restriction of some of its iterate gives a polinomial-like map of the same or lower degree. " [96]
  • parameter of the function

ParititionEdit

  • Markov

PeriodEdit

The smallest positive integer value p for which this equality

 

holds is the period[97] of the orbit.[98]

  is a point of periodic orbit ( limit cycle )  .

More is here

PerturbationEdit

PlaneEdit

Planes [101]

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

Dynamic planeEdit

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

Parameter planeEdit

See :[102]

Types of the parameter plane :

  • c-plane ( standard plane )
  • exponential plane ( map) [103][104]
  • flatten' the cardiod ( unroll ) [105][106] = "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)[107]
  • transformations [108]

PointsEdit

Band-mergingEdit

the band-merging points are Misiurewicz points[109]

BiaccessibleEdit

If there exist two distinct external rays landing at point we say that it is a biaccessible point.[110]

CenterEdit

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." [111]

Synonyms :

  • Nucleus of a Mu-Atom [112]

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." [113]

CriticalEdit

A critical point[114] of   is a point   in the dynamical plane such that the derivative vanishes:

 

Since

 

implies

 

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

  is an initial point for Mandelbrot set iteration.[115]

CutEdit

 
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).[116] 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.[117] Cut rays can be used to construct puzzles.

Cut angle is an angle of cut ray.

fixedEdit

Periodic point when period = 1

FeigenbaumEdit

 
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[118] 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)[119]

 

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[120] when the magnification increases by 4.6692 (the Feigenbaum Constant) and period is doubled each time[121]

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[122]

infinityEdit

The point at infinity [123]" 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[124]

MisiurewiczEdit

Misiurewicz point[125] = " parameters where the critical orbit is pre-periodic.

Examples are:

  • band-merging points of chaotic bands (the separator of the chaotic bands Bi−1 and Bi )[126]
  • the branch points
  • tips in the Mandelbrot set ( tips of the midgets ) [127]

Characteristic Misiurewicz pointof the chaotic band of the Mandelbrot set is :[128]

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

Myrberg-FeigenbaumEdit

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)[129]

PeriodicEdit

Point z has period p under f if :

 

In other words point is periodic

See also:

PinchingEdit

"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 [130]

post-criticalEdit

A post-critical point is a point

 

where   is a critical point.[131]

precriticalEdit

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

rootEdit

The root point :

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

singularEdit

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.

PortraitEdit

orbit portraitEdit

typesEdit

There are two types of orbit portraits: primitive and satellite.[132] 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.

PrecisionEdit

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."[133]
  • error [134]

Processes or transformations and phenomenonaEdit

Aliasing and antialisingEdit

Contraction and dilatationEdit

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

DiscretizationsEdit

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)[135]

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.[136]

NormalizationEdit

RenormalizationEdit

"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" [137]

TuningEdit

UniformizationEdit

VectorisationEdit

RadiusEdit

Radius of complex numberEdit

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

Conformal radiusEdit

Conformal radius of Siegel Disk [139][140]

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 :[141][142]

 

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[143]

SequencesEdit

A sequence is an ordered list of objects (or events).[144]

A series is the sum of the terms of a sequence of numbers.[145] Some times these names are not used as in above definitions.

OrbitEdit

Orbit can be:

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

SetEdit

ContinuumEdit

definition[146]

ComponentEdit

Components of parameter planeEdit

mu-atom , ball, bud, bulb, decoration, lake and lakelet.[147]

IslandsEdit

Names :

  • mini Mandelbrot set
  • 'baby'-Mandelbrot set
  • island mu-molecules = embedded copy of the Mandelbrot Set[148]
  • 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." [149]

  • primitive ( non-satellite)
    • the root of component is not on the boundaruy of another component
    • ones that have a cusp likes the main cardioid, when the the little Julia sets are disjoint [150]
  • satellite
    • ones that don't have a cusp[151]
    • it's root is on the boundary of another hyperbolic component [152]
    • when the little Julia sets touch at their β-fixed

point

primareEdit

Child (Descendant ) and the parent ( ancestor)Edit

  • ancestor of hyperbolic componnet
  • descendant of hyperbolic component = child [153]

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." [154]

A hyperbolic component is narrow if it contains no component of equal or lesser period in its wake [155]

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

LimbEdit

 
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.[156]

shrubEdit

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

WakeEdit

 
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."[157]

Components of dynamical planeEdit

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

dendritEdit

"Complex 1-variable polynomials with connected Julia sets and only repelling periodic points are called dendritic."[158]

DomainEdit

Domain in mathematical analysis it is an open connected set

Jordan domainEdit

"A Jordan domain[159] J is the the homeomorphic image of a closed disk in E2. The image of the boundary circle is a Jordan curve, which by the Jordan Curve Theorem separates the plane into two open domains, one bounded, the other not, such that the curve is the boundary of each." [160]

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). " [161] Other names:

  • level sets of integer escape time

FlowerEdit

Lea-Fatu flower

InvariantEdit

"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" [162]

Examples :

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

Level setEdit

in case of :

level set is defined :

 

Boundaries of level sets ( lemniscates) are

     

Planar setEdit

a non-separating planar set is a set whose complement in the plane is connected.[164]

SepalEdit

Sepal

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

TrapEdit

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

smoothEdit

smooth = changing without visible (noticeable) edges

use:

  • smooth gradient

similar:

  • conitnuous

compare:

  • discrete


SurgeryEdit

  • surgery in differential topology [165]

Links:

TestEdit

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.[166] It allows to find 2 sets :

  • escaping points ( it should be also the whole basing of attraction to infinity)[167]
  • 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).[168]

The output of test can be :

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

Attraction testEdit

ReferencesEdit

  1. Rational Maps with Clustering and the Mating of Polynomials by Thomas Joseph Sharland
  2. Topics from One-Dimensional Dynamics by Karen M. Brucks,Henk Bruin. page 265 exercise 14.2.12
  3. muency - internal angle ( the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2016. )
  4. internal angle from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2017
  5. argument of complex number
  6. A Method to Solve the Limitations in Drawing External Rays of the Mandelbrot Set M. Romera, G. Pastor, A. B. Orue, A. Martin, M.-F. Danca, and F. Montoya
  7. Matcont - is a Matlab software project for the numerical continuation and bifurcation study of continuous and discrete parameterized dynamical systems. Leaders of the project are Willy Govaerts (Gent,B) and Yuri A. Kuznetsov (Utrecht,NL).
  8. geometry by Dr. Carol JVF Burns
  9. What is a Curve  ?
  10. Unit circle in wikipedia
  11. The Road to Chaos is Filled with Polynomial Curves by Richard D. Neidinger and R. John Annen III. American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640-653
  12. Hao, Bailin (1989). Elementary Symbolic Dynamics and Chaos in Dissipative Systems. World Scientific. ISBN 9971-5-0682-3. http://power.itp.ac.cn/~hao/. 
  13. M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in one-dimensional quadratic maps", Physica A, 232 (1996), 517-535. Preprint
  14. Escape lines versus equipotential lines in the Mnadelbrot set by M. Romera, Pastor G , D. de la Guía, Montoya
  15. The Computation of Invariant Circles of Maps Article in Physica D Nonlinear Phenomena 16(2):243-251 · June 1985 DOI: 10.1016/0167-2789(85)90061-2 1st I.G. Kevrekidis
  16. A Newton-Raphson method for numerically constructing invariant curves Marty, Wolfgang
  17. Numerical Approximation of Rough Invariant Curves of Planar Maps Article in SIAM Journal on Scientific Computing 25(1) · September 2003 DOI: 10.1137/S106482750241373X K. D. Edoh and Jens Lorenz
  18. SIAM J. Sci. and Stat. Comput., 8(6), 951–962. (12 pages) A New Algorithm for the Numerical Approximation of an Invariant Curve Published online: 14 July 2006 Keywords invariant manifold, polygonal approximation AMS Subject Headings 65L99, 65H10, 34C40 Publication Data ISSN (print): 0196-5204 ISSN (online): 2168-3417 Publisher: Society for Industrial and Applied Mathematics M. van Veldhuizen
  19. Escape lines versus equipotential lines in the Mnadelbrot set by M. Romera, Pastor G , D. de la Guía, Montoya
  20. wikipedia : Jordan curve theorem
  21. Modeling Julia Sets with Laminations: An Alternative Definition by Debra Mimbs
  22. Laminations of the unit disk with irrational rotation gaps by John C. Mayer
  23. Rational maps represented by both rabbit and aeroplane matings Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Freddie R. Exall July 2010
  24. Rational maps represented by both rabbit and aeroplane matings Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Freddie R. Exall July 2010
  25. Iterated Monodromy Groups of Quadratic Polynomials, I Laurent Bartholdi, Volodymyr V. Nekrashevych
  26. GROWTH OF GROUPS DEFINED BY AUTOMATA : ASHLEY S. DOUGHERTY, LYDIA R. KINDELIN, AARON M. REAVES, ANDREW J. WALKER, AND NATHANIEL F. ZAKAHI
  27. wikipedi : dense_set
  28. mathoverflow question : is-there-an-almost-dense-set-of-quadratic-polynomials-which-is-not-in-the-inte/254533#254533
  29. fractalforums : dense-image
  30. A Cheritat wiki : see image showing gamma-correct downscale of dense part of Mandelbropt set
  31. fractal forums : pathfinding-in-the-mandelbrot-set/
  32. A Cheritat wiki : Mandelbrot_set - Following_the_derivative
  33. wikipedia : discretization
  34. mathinsight : from_discrete_to_continuous_dynamical_systems
  35. Symbolic Dynamics of Quadratic Polynomials by H. Bruin and D. Schleicher
  36. Symbolic Dynamics and Rotation Numbers J. J. P. Veerman Phys. 13A, 1986, 543-576.
  37. Symbolic Dynamics of Order-Preserving Orbits J. J. P. Veerman Phys. 29D, 1987, 191-201.
  38. Walter Bergweiler : A gallery of complex dynamics pictures.
  39. mathoverflow question : whats-a-natural-candidate-for-an-analytic-function-that-interpolates-the-tower/43003
  40. A Cheritat wiki  : Mandelbrot_set - Interior_detection_methods
  41. MAT335H1F Lecture Notes by Burbulla (Chapter 11, 12 and 13 )
  42. Germ in wikipedia
  43. math.stackexchange question : is-there-any-difference-between-mapping-and-function
  44. Iterated function (map) in wikipedia
  45. evolution function
  46. the discrete nonlinear dynamical system
  47. Connectivity of Julia sets of Newton maps: A unified approach by K. Baranski N. Fagella X. Jarque B. Karpinska
  48. A Beginners’ Guide to Resurgence and Trans-series in Quantum Theories Gerald Dunne
  49. dinkydauset at deviantar :Perturbation-for-the-Mandelbrot-set-450766847
  50. Dessins d’enfants and Hubbard trees by Kevin M. Pilgrim
  51. 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.
  52. math.stackexchange question : definition-of-markov-partition
  53. Structure of Inverse Limit Spaces of Tent Maps with Nonrecurrent Critical Points by Brian Raines and Sonja Stimac
  54. Bifurcation structures in maps of Henon type by Kai T Hansen and Predrag Cvitanovic
  55. wikipedia: Magnitude in mathematics
  56. Hyperbolic Components by John Milnor
  57. Complex quadratic map in wikipedia
  58. Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
  59. Mandel: software for real and complex dynamics by Wolf Jung
  60. three-cool-facts-about-rotations-of-the-circle by David Richeson
  61. irrational-rotations-of-the-circle-and-benfords-law by David Richeson
  62. wikipedia : Dyadic transformation
  63. lavaurs' algorithm in Haskell with SVG output by Claude Heiland-Allen
  64. SYMBOLIC DYNAMICS AND SELF-SIMILAR GROUPS by VOLODYMYR NEKRASHEVYCH
  65. The measure of the Feigenbaum Julia set by Artem Dudko and Scott Sutherland
  66. Poincaré map
  67. General principles of chaotic dynamics by P.B. Persson , C.D. Wagner
  68. Continuous time and discrete time dynamical systems by Shaun Bullett
  69. Continuous time and discrete time dynamical systems by Shaun Bullett
  70. Conformal Geometry and Dynamics of Quadratic Polynomials Mikhail Lyubich
  71. A THOMPSON GROUP FOR THE BASILICA by JAMES BELK AND BRADLEY FORREST
  72. math stackexchange question: explicit-riemann-mappings
  73. mathoverflow question: complex-function-for-mapping-a-circle-to-a-superellipse
  74. math.stackexchange question: explicit-riemann-mappings
  75. binary_shift_left
  76. Multiplier at wikipedia
  77. Internal angles and multipliers from Fractal Geometry Yale University Michael Frame, Benoit Mandelbrot (1924-2010), and Nial Neger September 3, 2017
  78. A Cheritat wiki-draw : Mandelbrot_set#Following_the_derivative
  79. wikipedia : Rotation number
  80. rotation number From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2016
  81. scholarpedia : Rotation_theory
  82. The Fractal Geometry of the Mandelbrot Set II. How to Count and How to Add Robert L. Devaney
  83. An Introduction to Rotation Theory Prize winner, DSWeb Student Competition, 2007 By Christian Kue
  84. Complex systems simulation Curso 2012-2013 by Antonio Giraldo and María Asunción Sastre
  85. https://plus.maths.org/content/winding-numbers-topography-and-topology-ii
  86. wikipedia : Orbit_(dynamics)
  87. wikipedia : orbit (dynamics)
  88. Wikipedia : Complex quadratic polynomial - Critical point
  89. MandelOrbits - A visual real-time trace of Mandelbrot iterations by Ivan Freyman
  90. wikipedia : Periodic points of complex quadratic mappings
  91. M. Romera, G. Pastor, and F. Montoya : Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Fractalia 6, No. 21, 10-12 (1997)
  92. 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
  93. Khan Academy : Mandelbrot Spirals 2
  94. Complex Power Towers (Or ‘mucking around with Mathematica’) by Mike Croucher
  95. /DarkHeart by Chris King
  96. Ouadratic-like maps and Renormalization by Nuria Fagella
  97. Peiod From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015.
  98. scholarpedia : Periodic Orbit for a Map
  99. Perturbation for the Mandelbrot set by DinkydauSet, Apr 28, 2014, 3:46:13 PM
  100. PARABOLIC IMPLOSION A MINI-COURSE ARNAUD CHERITAT
  101. wikipedia : Complex_quadratic_polynomial - Planes
  102. Alternate Parameter Planes by David E. Joyce
  103. mu-ency : exponential map by R Munafo
  104. Exponential mapping and OpenMP by Claude Heiland-Allen
  105. Linas Vepstas : Self Similar?
  106. the flattened cardioid of a Mandelbrot by Tom Rathborne
  107. Stretching cusps by Claude Heiland-Allen
  108. Twisted Mandelbrot Sets by Eric C. Hill
  109. doubling bifurcations on complex plane by E Demidov
  110. On biaccessible points in the Julia set of the family z(a+z^{d}) by Mitsuhiko Imada
  111. Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper
  112. Nucleus - From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2015.
  113. Siegel disks by Xavier Buff and Arnaud Ch ́ritat e Univ. Toulouse Roma, April 2009
  114. wikipedia : Critical point (mathematics)
  115. Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations
  116. Cut point in wikipedia
  117. On local connectivity for the Julia set of rational maps : Newton’s famous example By P. Roesch
  118. muency : feigenbaum point
  119. On Periodic and Chaotic Regions in the Mandelbrot Set by G. Pastor , M. Romera, G. Álvarez, D. Arroyo and F. Montoya
  120. fractal-faq : section 6
  121. Period doubling and Feigenbaum's scaling be E Demidov
  122. mathoverflow question : is-there-a-way-to-find-regions-of-depth-in-the-mandelbrot-set-other-than-simply?rq=1
  123. [w:Point at infinity|Point at infinity in wikipedia]
  124. Mathoverflow question : Attractive Basins and Loops in Julia Sets
  125. wikipedia : Misiurewicz point
  126. Symbolic sequences of one-dimensional quadratic map points by G Pastor, Miguel Romera, Fausto Montoya Vitini
  127. mathoverflow question : Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?
  128. G. Pastor, M. Romera, G. Álvarez, D. Arroyo and F. Montoya, "On periodic and chaotic regions in the Mandelbrot set", Chaos, Solitons & Fractals, 32 (2007) 15-25
  129. The bifurcation diagram of cubic polynomial vector fields on CP1 by Christiane Rousseau
  130. http://www.mndynamics.com/indexp.html%7C program Mandel by Wolf Jung , demo 2 page 3
  131. GROWTH OF GROUPS DEFINED BY AUTOMATA : ASHLEY S. DOUGHERTY, LYDIA R. KINDELIN, AARON M. REAVES, ANDREW J. WALKER, AND NATHANIEL F. ZAKAHI
  132. wikipedia : Orbit portrait
  133. stackoverflow question : how-do-i-interpret-precision-and-scale-of-a-number-in-a-database
  134. taking-the-error-out-of-the-error-function by Fredrik Johansson
  135. Airplane primitive parabolic implosion by Wolf Jung
  136. CANTOR BOUQUETS, EXPLOSIONS, AND KNASTER CONTINUA: DYNAMICS OF COMPLEX EXPONENTIALS by Robert L. Devaney Publicacions Matematiques, Vol 43 (1999), 27–54.
  137. Baby Mandelbrot sets, Renormalization and MLC Mikhail Lyubich
  138. Tuning From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2014.
  139. wikipedia : Conformal radius
  140. scholarpedia : Quadratic Siegel disks
  141. Julia Sets of Complex Polynomials and Their Implementation on the Computer by Christoph Martin Stroh
  142. fractalforums: bounding circle of julia sets by knighty
  143. The Mandelbrot Set and Julia Sets Combinatorics in the Mandelbrot Set - The 1/n2 rule, and deviations from it
  144. wikipedia : Sequence
  145. wikipedia : series
  146. wikipedia : Continuum in set theory
  147. Mu-atom From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2013.
  148. Island Mu-Molecule by Robert P. Munafo, 2012 Aug 18.
  149. Internal addresses in the Mandelbrot set and Galois groups of polynomials by Dierk Schleicher, page 31
  150. Satellite copies of the Mandelbrot set by Luna Lomonaco
  151. mathoverflow : precise-location-of-the-mandelbrot-bulb-attachment-to-the-main-cardioid
  152. ON BIACCESSIBLE POINTS OF THE MANDELBROT SET by SAEED ZAKERI
  153. Child From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2013.
  154. Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper
  155. Internal addresses in the Mandelbrot set and irreducibility of polynomials by Dierk Schleicher
  156. Gary William Flake, The Computational Beauty of Nature, 1998. p. 125. ISBN 978-0-262-56127-3.
  157. Local properties of the Mandelbrot set at parabolic points by Tan Lei
  158. LAMINATIONAL MODELS FOR SOME SPACES OF POLYNOMIALS OF ARBITRARY DEGREE by ALEXANDER BLOKH, LEX OVERSTEEGEN, ROSS PTACEK, AND VLADLEN TIMORIN
  159. wikipedia : Carathéodory's theorem (conformal mapping)
  160. The intrinsic geometry of a Jordan domain by Richard L. Bishop
  161. fractalforums : binary-decomposition-and-external-angles by Claude
  162. Norman Lebovitz : Textbook for Mathematics 27300
  163. wikipedia : Level set
  164. 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
  165. wikipedioa: Surgery theory
  166. Fractus doc by Richard Rosenman
  167. wikipedia : Escaping set
  168. fractint doc : bailout