angled

Angled internal address is an extension of internal address

 ${\displaystyle 1\quad {\xrightarrow {1/3}}\ 3\quad {\xrightarrow {1/2}}\ 6}$

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

Problems

• islands
• infinite sequence of bifurcations

external

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

internal

The internal angle^[5] 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 ( inside child component)
• "The internal angles start at 0, at the cusp, and increase counterclockwise. " Robert Munafo^[6]

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

plain

The plain angle is an agle of complex point = it's argument ^[7]

Circle

Equipotential lines

Equipotential lines = Isocurves of complex potential

"If the escape radius is greater than 2 the contour lines are equipotential lines" ^[25]

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^[33]

External ray

Internal ray

Definition:

• "The internal rays are the preimages of the radial segments under the coordinate with componenet center corresponding to 0." Alfredo Poirier^[34]

Internal rays are :

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

Spider

A spider S is a collection of disjoint simple curves called legs ^[38]( extended rays = external + internal ray) in the complex plane connecting each of the post-critical points to infnity ^[39]

See :

• spider algorithm
• spider program

Derivative with respect to c

On parameter plane :

• ${\displaystyle c}$ is a variable
• ${\displaystyle z_{0}=0}$ is constant

${\displaystyle {\frac {d}{dc}}f_{c}^{(p)}(z_{0})=z'_{p}\,}$

This derivative can be found by iteration starting with

${\displaystyle z_{0}=0\,}$

${\displaystyle z'_{0}=1\,}$

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

${\displaystyle z'_{p}=2\cdot z_{p-1}\cdot z'_{p-1}+1\,}$

${\displaystyle z_{p}=z_{p-1}^{2}+c\,}$

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

${\displaystyle {\frac {dz_{n+1}}{dc}}={\frac {d(z_{n}^{2}+c)}{dc}}=2z_{n}{\frac {dz_{n}}{dc}}+{\frac {dc}{dc}}=2z_{n}{\frac {dz_{n}}{dc}}+1\,}$

• 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 :

${\displaystyle z_{0}=0\qquad \qquad z'_{0}=1\,}$

${\displaystyle z_{1}=c\qquad \qquad z'_{1}=1\,}$

${\displaystyle z_{2}=c^{2}+c\qquad z'_{2}=2c+1\,}$

It can be used for:

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

Derivative with respect to z

${\displaystyle z'_{n}\,}$ is first derivative with respect to z.

This derivative can be found by iteration starting with

${\displaystyle z'_{0}=1\,}$

and then :

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

description	arbitrary name	formula	Initial conditions	Recurrence step	common names
iterated complex quadratic polynomial	${\displaystyle A_{m}}$	${\displaystyle f^{m}}$	${\displaystyle A_{0}=z}$	${\displaystyle A_{m+1}=A_{m}^{2}+c}$	z or f
first derivative with respect to z	${\displaystyle B_{m}}$	${\displaystyle {\dfrac {\partial }{\partial z}}f^{m}}$	${\displaystyle B_{0}=1}$	${\displaystyle B_{m+1}=2A_{m}B_{m}}$	dz, d (from derivative) or p ( from prime) of f'

Derivation of recurrence relation:

${\displaystyle A_{0}\quad {\xrightarrow {definition\ of\ A}}\ f^{0}(z,c)\quad {\xrightarrow {definition\ of\ f}}\ \quad z}$

${\displaystyle A_{m+1}\quad {\xrightarrow {definitionofA}}f^{m+1}(z,c)\quad {\xrightarrow {definition\ of\ f}}f(f^{m}(z,c),c)\quad {\xrightarrow {definition\ of\ A}}f(A_{m},c)\quad {\xrightarrow {definition\ of\ f}}A_{m}^{2}+c}$

${\displaystyle B_{0}\quad {\xrightarrow {definition\ of\ B}}\ {\dfrac {\partial }{\partial z}}f^{0}(z,c){\xrightarrow {definition\ of\ f}}\ {\dfrac {\partial }{\partial z}}z{\xrightarrow {derivative}}\ 1}$

${\displaystyle B_{m+1}\quad {\xrightarrow {definition\ of\ B}}{\dfrac {\partial }{\partial z}}A_{m+1}\quad {\xrightarrow {definition\ of\ A}}{\dfrac {\partial }{\partial z}}(A_{m}^{2}+c)\quad {\xrightarrow {distributivity}}{\dfrac {\partial }{\partial z}}A_{m}^{2}+{\dfrac {\partial }{\partial z}}c\quad {\xrightarrow {constant\ derivative}}{\dfrac {\partial }{\partial z}}A_{m}^{2}+0\quad {\xrightarrow {zero}}{\dfrac {\partial }{\partial z}}A_{m}^{2}\quad {\xrightarrow {chain\ rule}}2A_{m}({\dfrac {\partial }{\partial z}}A_{m})\quad {\xrightarrow {definition\ of\ B}}2A_{m}B_{m}}$

It can be used for :

• computing the external distance to the Julia set ( DEM/J )
• detection of interior points of Mandelbrot set componnets^[54]

unsigned char ComputeColorOfDEMJ(complex double z){
// https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Julia_set#DEM.2FJ


  int nMax = iterMax;
  complex double dz = 1.0; //  is first derivative with respect to z.
  double distance;
  double cabsz;
	
  int n;

  for (n=0; n < nMax; n++){ //forward iteration

    if (cabs(z)> 1e60 || cabs(dz)> 1e60) break; // big values
    
  			
    dz = 2.0*z * dz; 
    z = z*z +c ; /* forward iteration : complex quadratic polynomial */ 
  }
  
  cabsz = cabs(z);
  distance = 2.0 * cabsz* log(cabsz)/ cabs(dz);
  if (distance <distanceMax) {//printf(" distance = %f \n", distance); 
  		return iColorOfBoundary;}
  
  
  return iColorOfExterior;
}

The Schwarzian Derivative

The Schwarzian Derivative ^{[55] [56][57][58]}

Wirtinger derivatives

gradient

the gradient is the generalization of the derivative for the multivariable functions^[59][60]

definitions:

• (field): Gradient field is the vector field with gradient vector
• (function): The gradient of a scalar-valued multivariable function ${\displaystyle f(x,y)}$ is a vector-valued function denoted ${\displaystyle \nabla f}$
• (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^[61]
• (operator): Del or nabla is an gradient operator = a vector differential operator

• 
  Gradient 2D field with gradient vectors
• 
  one gradient vector (red)

Notations:

${\displaystyle \nabla f(x,y)=[{\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}}]={\begin{bmatrix}{\frac {\partial f}{\partial x}}\\{\frac {\partial f}{\partial y}}\end{bmatrix}}={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} }$

${\displaystyle \operatorname {grad} f=\nabla f}$

See also

• Gradient Descent Algorithm^[62][63]
• Gradient Ascent Algorithm
• image gradient

Jacobian

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

Brjuno

• Brjuno function

Links:

• Quelle est la régularité de la fonction de Brjuno ?

harmonic

An harmonic or spherical function is a:

• "set of orthogonal functions all of whose curvatures are changing at the same rate."^[70]
• "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"^[71]

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 ? ^[72]

meromorphic

meromorphic maps: Those with NO FINITE, NON-ATTRACTING FIXED POINTS^[73]

Polynomial

Resurgent

"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^[74][75][76]

Yoccoz’s function

Farey tree

Farey tree = Farey sequence as a tree

Hubbard tree

• 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." ^[80]
• " 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^[81]

Rooted tree

rooted tree of preimages:

${\displaystyle T_{f}:=\bigsqcup _{n\geq 0}f^{-n}(t),}$

where a vertex ${\displaystyle z\in f^{-n}(t)}$ is connected by an edge with ${\displaystyle f(z)\in f^{-(n-1)}(t)}$.

Complex quadratic map

Forms

c form : ${\displaystyle z^{2}+c}$

quadratic map^[84]

• math notation : ${\displaystyle f_{c}(z)=z^{2}+c\,}$
• 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

${\displaystyle \ f_{c}^{(0)}(z)=z=z_{0}}$

${\displaystyle \ f_{c}^{(1)}(z)=f_{c}(z)=z_{1}}$

...

${\displaystyle \ f_{c}^{(p)}(z)=f_{c}(f_{c}^{(p-1)}(z))}$

or with subscripts :

${\displaystyle \ z_{p}=f_{c}^{(p)}(z_{0})}$

• Maxima CAS function :

fn(p, z, c) :=
  if p=0 then z
  elseif p=1 then f(z,c)
  else f(fn(p-1, z, c),c);

zp:fn(p, z, c);

lambda form : ${\displaystyle z^{2}+\lambda z}$

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

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

Switching between forms

Start from :

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

Multiplier of fixed point :

${\displaystyle \lambda =re^{2\pi \theta i}}$

When one wants change from lambda to c :^[85]

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

or from c to lambda :

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

Example values :

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

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

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

of period 1 hyperbolic component ( main cardioid) for given internal angle ( rotation number) t using this c / cpp code by Wolf Jung^[86]

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 map

Circle map ^[87]

• irrational rotation^[88]

Doubling map

definition ^[89][90][91]

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^[92]

-- 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 map

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

• one image = 2α mod 1 under doubling map
• "two preimages under the doubling map: α/2 and (α + 1)/2.".^[93] Inverse of doubling map is multivalued function.

In Maxima CAS :

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

Note that difference between these 2 preimages

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

is half a turn = 180 degrees = Pi radians.

Images and preimages under doubling map d

${\displaystyle \alpha }$	${\displaystyle d^{1}(\alpha )}$	${\displaystyle d^{-1}(\alpha )}$
${\displaystyle {\frac {1}{2}}}$	${\displaystyle {\frac {1}{1}}}$	${\displaystyle \left\{{\frac {1}{4}},{\frac {3}{4}}\right\}}$
${\displaystyle {\frac {1}{3}}}$	${\displaystyle {\frac {2}{3}}}$	${\displaystyle \left\{{\frac {1}{6}},{\frac {4}{6}}\right\}}$
${\displaystyle {\frac {1}{4}}}$	${\displaystyle {\frac {1}{2}}}$	${\displaystyle \left\{{\frac {1}{8}},{\frac {5}{8}}\right\}}$
${\displaystyle {\frac {1}{5}}}$	${\displaystyle {\frac {2}{5}}}$	${\displaystyle \left\{{\frac {1}{10}},{\frac {6}{10}}\right\}}$
${\displaystyle {\frac {1}{6}}}$	${\displaystyle {\frac {1}{3}}}$	${\displaystyle \left\{{\frac {1}{12}},{\frac {7}{12}}\right\}}$
${\displaystyle {\frac {1}{7}}}$	${\displaystyle {\frac {2}{7}}}$	${\displaystyle \left\{{\frac {1}{14}},{\frac {4}{7}}\right\}}$

Feigenbaum map

"the Feigenbaum map F is a solution of Cvitanovic-Feigenbaum equation"^[94]

First return map

• definition ^[95]
• video : Intro to Poincare map (Poincaré), the first return map. This map helps us determine the stability of a limit cycle using the eigenvalues (Floquet multipliers) associated with the map.

"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 "^[96]

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

"The first return map S → S is the map defined by sending each x0 ∈ S to the point of S where the orbit of x0 under the system first returns to S." ^[97]

"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"^[98]

postcritically finite

postcritically finite: maps whose critical orbits are all periodic or preperiodic^[99]

  " 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:^[100]

• 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)

Multiplier map

Mandelbrot set - multiplier map

Multiplier map ${\displaystyle \lambda }$ associated with hyperbolic component ${\displaystyle \mathrm {H} }$

• gives an explicit uniformization of hyperbolic component ${\displaystyle \mathrm {H} }$ by the unit disk ${\displaystyle \mathbb {D} }$ :

${\displaystyle \lambda _{p}:\mathrm {H} \to \mathbb {D} }$

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:

${\displaystyle \lambda _{p}(c)=b}$

where:

• c is a point in the parameter plane
• b is a point in the reference plane. It is also internal coordinate
• ${\displaystyle \lambda }$ is a multiplier map

Multiplier map is a conformal isomorphism.^[101]

It can be computed using :

• Newton_method : internal_coordinate by Claude Heiland-Allen
• m-interior function

Riemann map

Riemann mapping theorem^[102] says that every simply connected subset U of the complex number plane can be mapped to the open unit disk D

${\displaystyle f:U\to D}$

where:

• D is a unit disk ${\displaystyle D=\{z\in \mathbf {C} :|z|<1\}.}$
• 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^[103]
    • for the Fatou component containing a superattracting fixed point for a rational map^[104]
    • 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)^[105]
  • Zipper
    • https://code.google.com/archive/p/zipper/
    • https://sites.math.washington.edu/~marshall/zipper.html
  • " 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^[106]

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

 ${\displaystyle F(z)\;=\;-\log |z|}$

for all ${\displaystyle z\in \partial D}$, and let

 ${\displaystyle R(z)=|z|e^{F(z)}}$

Then

• ${\displaystyle R(0)=0}$
• ${\displaystyle R|_{\partial D}\equiv 1}$
• and ${\displaystyle \log R}$ 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 map

     "If a is rational, then every point is periodic. If a is irrational, then every point has a dense orbit." David Richeson[107]

 ${\displaystyle R_{\frac {p}{q}}(\theta )=\theta +{\frac {p}{q}}}$

Shift map

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

 ${\displaystyle \sigma (b_{1},b_{2},b_{3},\ldots )=(b_{2},b_{3},b_{4},\ldots )}$

It just drops first digit of the sequence.

  ${\displaystyle \sigma ^{2}(S)=\sigma (\sigma (S))}$

  ${\displaystyle \sigma ^{k}(b_{1}b_{2}\ldots )=b_{k+1}b_{k+2}\ldots }$

If we treat sequence as a binary fraction :

 ${\displaystyle x=0.b_{1},b_{2},b_{3},\ldots }$

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

 ${\displaystyle \sigma (x)=2x{\bmod {1}}}$

and "shifting N places left is the same as multiplying by 2 to the power N (written as 2N)"^[108] ( operator << )

In Haskell:

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

See also:

• How to compute external angles of principal Misiurewicz points of wakes?

Fegenbaum constant

• first ( delta)^[113]
• second ( alpha)

How to compute:

• Keith Briggs: How to calculate
• octave program by Anton Hendricson
• python program by cdlane
• Rosettacode
• An efficient method for the computation of the Feigenbaum constants to high precision by Andrea Molteni (Submitted on 7 Feb 2016)

Kneading paritition of the dynamic plane

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:

${\displaystyle z=c}$

• 
  t = 1/4 preperiod = 2 period = 1
• 
  t = 1/6 preperiod = 1 and period = 2
• 
  t = 9/56 preperiod = 3 and period = 3

Spine paritition of the dynamic plane

Curve ${\displaystyle R\,}$ :

${\displaystyle R\ {\overset {\underset {\mathrm {def} }{}}{=}}\ R_{1/2}\ \cup \ S_{c}\ \cup \ R_{0}\,}$

where:

• R is an dynamic external ray
• S is the spine of Julia set

divides dynamical plane into two components.

• 
  Plane paritition in case of Rabbit Julia set
• 
  Plane paritition in case of Basilica Julia set

crossing/noncrossing

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)^[143]

Key words:

• Enumerative combinatorics

types in case of discrete dynamical system

Nucleus or center of hyperbolic component

A center of a hyperbolic component H is a parameter ${\displaystyle c_{0}\in H\,}$ ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." ^[155]

Synonyms :

• Nucleus of a Mu-Atom ^[156]

How to find center/s ?

Center of Siegel Disc

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." ^[157]

infinitely renormalizable

" 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^[179]

types

There are two types of orbit portraits: primitive and satellite.^[181] If ${\displaystyle v}$ is the valence of an orbit portrait ${\displaystyle {\mathcal {P}}}$ and ${\displaystyle r}$ is the recurrent ray period, then these two types may be characterized as follows:

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