Fractals/Iterations in the complex plane/def cqp

< Fractals < Iterations in the complex plane

Definitions

Address

Internal

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

Internal address :

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

angled

Angled internal address is an extension of internal address

Angle

Types of angle

Principal branch or complex number argument

	external angle	internal angle	plain angle
parameter plane	$arg(\Phi _{M}(c))\,$	$arg(\rho _{n}(c))\,$	$arg(c)\,$
dynamic plane	$arg(\Phi _{c}(z))\,$		$arg(z)\,$

where :

$\rho _{n}(c)$ is a multiplier map
$\Phi (c)\,$ is a Boettcher function

external

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

internal

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

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

plain

The plain angle is an agle of complex point = it's argument ^[4]

Units

turns
degrees
radians

Number types

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 :^[5]

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

where :

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

Bifurcation

Numerical Bifurcation Analysis of Maps
- MatCont^[6]

Coordinate

Coordinate :

Fatou coordinate for every repelling and attracting petal ( linearization of function near parabolic fixed point )
Boettcher
Koenigs

Curves

Types:

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

Description^[7]

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

closed

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.^[8]
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"

Circle

Unit circle

Unit circle $\partial D\,$ is a boundary of unit disk^[9]

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

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

$w=e^{i*t}\,$

Critical curves

Diagrams of critical polynomials are called critical curves.^[10]

These curves create skeleton of bifurcation diagram.^[11] (the dark lines^[12])

Escape lines

"If the escape radius is equal to 2 the contour lines have a contact point (c= -2) and cannot be considered as equipotential lines" ^[13]

Invariant

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^[14]^[15]^[16]
curve is invariant for a system of ordinary differential equations^[17]

Isocurves

Equipotential lines

Equipotential lines = Isocurves of complex potential

"If the escape radius is greater than 2 the contour lines are equipotential lines" ^[18]

Jordan curve

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^[19]

Lamination

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^[20]^[21]

It is a model of Mandelbrot or Julia set.

A lamination, L, is a union of leaves and the unit circle which satisfies :^[22]

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

Leaf

Chords = leaves = arcs

A leaf on the unit disc is a path connecting two points on the unit circle. ^[23]

Open curve

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

Ray

Rays are :

invariant curves
dynamic or parameter
external or internal

Internal ray

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 )

Spider

A spider S is a collection of disjoint simple curves called legs ^[24]( extended rays = external + internal ray) in the complex plane connecting each of the post-critical points to infnity ^[25]

See :

Vein

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

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

Discretization

discretization^[26] and its reverse ^[27]

Dynamics

symbolic^[28] ^[29]^[30]
complex
Arithmetic
combinatorial

symbolic

"Symbolic dynamics encodes :

a dynamical system $f:X\to X$ by a shift map on a space of sequences over finite alphabet using Markov paritition of the space $X$
the points of space $X$ by their itineraries with respect to the paritition " ( Volodymyr Nekrashevych - Symbolic dynamics and self-similar groups

equation

differential

differential equations

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

Function

Derivative

Derivative of Iterated function (map)

Derivative with respect to c

On parameter plane :

$c$ is a variable
$z_{0}=0$ is constant

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

This derivative can be found by iteration starting with

z_{0}=0\,

z'_{0}=1\,

and then

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

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

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

Maxima CAS function :

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

Example values :

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

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

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

Derivative with respect to z

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

This derivative can be found by iteration starting with

z'_{0}=1\,

and then :

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

Germ

Germ ^[31] of the function f in the neighborhood of point z is a set of the functions g which are indistinguishable in that neighborhood

[f]_{z}=\{g:g\sim _{z}f\}.

See :

parabolic germ

map

differences between map and the function ^[32]
Iterated function = map^[33]
an evolution function^[34] of the discrete nonlinear dynamical system^[35]

z_{n+1}=f_{c}(z_{n})\,

is called map $f_{c}$ :

f_{c}:z\to z^{2}+c.\,

types

Polynomial

Critical

Critical polynomial :

$Q_{n}=f_{c}^{n}(z_{cr})=f_{c}^{n}(0)\,$

$Q_{1}=f_{c}^{1}(0)=c\,$

$Q_{2}=f_{c}^{2}(0)=c^{2}+c\,$

$Q_{3}=f_{c}^{3}(0)=(c^{2}+c)^{2}+c\,$

These polynomials are used for finding :

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

post-critically finite

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

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^[36]

glitches

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

Invariants

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

Dynamical invariants = invariants of the dynamical system

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

Dynamical Invariants Derived from Recurrence Plots^[38]

Interval

a partition of an interval into subintervals

Markov paritition^[39]

Iteration

Itinerary

$S(x)$ is an itinerary of point x under the map f relative to the paritirtion.

It is a right-infinite sequence of zeros and ones ^[40]

  $S(x)=s_{1}s_{2}s_{3}...s_{n}$

where

Examples :

internal angle and wake

For rotation map $R_{p/q}$ and invariant interval $I$ ( circle ) :

 $I=(0,1]$

one can compute $x_{c}$ :

  $x_{c}=1-{\frac {p}{q}}$

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

 $I_{0}=(0,x_{c}]$ 

 $I_{1}=(x_{c},1]$

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

$s_{n}={\begin{cases}0:x_{n}<x_{c}\\1:x_{n}>x_{c}\end{cases}}$

Itinerary can be converted^[41] to point $x\in [0,1]$

 $\gamma (S_{n})=0.s_{1}s_{2}s_{3}...s_{n}=\sum _{n=0}^{n-1}{\frac {s_{n}}{2^{n}}}=x_{n}$

Magnitude

magnitude of the point ( complex number in 2D case) = it's distance from the origin

Map

description

types

The map f is hyperbolic if every critical orbit converges to a periodic orbit.^[42]

Complex quadratic map

Forms

c form : $z^{2}+c$

quadratic map^[43]

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

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

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

...

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

or with subscripts :

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

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

Multiplier of fixed point :

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

When one wants change from lambda to c :^[44]

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

or from c to lambda :

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

Example values :

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

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

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

of period 1 hyperbolic component ( main cardioid) for given internal angle ( rotation number) t using this c / cpp code by Wolf Jung^[45]


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 ^[46]

irrational rotation^[47]

Doubling map

definition ^[48]

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^[49]

-- by Claude Heiland-Allen
-- type Q = Rational
 double :: Q -> Q
 double p
   | q >= 1 = q - 1
   | otherwise = q
   where q = 2 * p

//  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." ^[50]. Inverse of doubling map is multivalued function.

In Maxima CAS :

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

Note that difference between these 2 preimages

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

is half a turn = 180 degrees = Pi radians.

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

Feigenbaum map

"the Feigenbaum map F is a solution of Cvitanovic-Feigenbaum equation"^[51]

First return map

definition ^[52]

"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 "^[53]

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

"The first return map S → S is the map defined by sending each x0 ∈ S to the point of S where the orbit of x0 under the system first returns to S." ^[54]

"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"^[55]

Multiplier map

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

$\lambda :\mathrm {H} \to \mathbb {D}$

Multiplier map is a conformal isomorphism.^[56]

Rotation map

Rotation map $R$ describes counterclockwise rotation of point $\theta$ thru ${\frac {p}{q}}$ turns on the unit circle :

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

It is used for computing :

itinerary

rotation maps

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

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

It just drops first digit of the sequence.

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

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

If we treat sequence as a binary fraction :

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

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

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

and "shifting N places left is the same as multiplying by 2 to the power N (written as 2N)"^[57] ( operator << )

Multiplier

Multiplier of periodic z-point : ^[58]

Math notation :

$\lambda _{c}(z)={\frac {df_{c}^{(p)}(z)}{dz}}\,$

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	$f^{p}(z)\,$	$\lambda _{c}(z)\,$
1	$z^{2}+c\,$	$2z\,$
2	$z^{4}+2cz^{2}+c^{2}+c$	$4z^{3}+4cz$
3	$z^{8}+4cz^{6}+6c^{2}z^{4}+2cz^{4}+4c^{3}z^{2}+4c^{2}z^{2}+c^{4}+2c^{3}+c^{2}+c$	$8z^{7}+24cz^{5}+24c^{2}z^{3}+8cz^{3}+8c^{3}z+8c^{2}z$

It is used to :

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

Number

Rotation number

The rotation number^[59]^[60]^[61]^[62]^[63] 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 ^[64]

Features :

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

Winding number

def^[65]

Orbit

Orbit is a sequence of points = trajectory

Critical

Forward orbit^[66] of a critical point^[67]^[68] is called a critical orbit. Critical orbits are very important because every attracting periodic orbit^[69] attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.^[70]^[71] ^[72]

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

$z_{1}=f_{c}(z_{0})=c\,$

$z_{2}=f_{c}(z_{1})=c^{2}+c\,$

$z_{3}=f_{c}(z_{2})=(c^{2}+c)^{2}+c\,$

$...\,$

This orbit falls into an attracting periodic cycle.

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:

images of critical orbitsat commons
by Mike Croucher^[73]
Chris King ^[74]
list : critical orbits
images by Conan written in Rainbow
- intro
- examples

Forward

Homoclinic / heteroclinic

homoclinic orbit
Heteroclinic orbit

Inverse

Inverse = Backward

Parameter

point of parameter plane : " is renormalizable if restriction of some of its iterate gives a polinomial-like map of the same or lower degree. " ^[75]
parameter of the function

Paritition

Markov

Period

The smallest positive integer value p for which this equality

 $f^{p}(z_{0})=z_{0}$

holds is the period^[76] of the orbit.^[77]

$z_{0}$ is a point of periodic orbit ( limit cycle ) $\{z_{0},\dots ,z_{p-1}\}$ .

More is here

Perturbation

Perturbation technque for fast rendering the deep zoom images of the Mandelbrot set^[78]
perturbation of parabolic point ^[79]
use perturbation theory to approximate the solutions of the differential equations

Plane

Planes ^[80]

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

Dynamic plane

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

Parameter plane

See :^[81]

Types of the parameter plane :

c-plane ( standard plane )
exponential plane ( map) ^[82]^[83]
flatten' the cardiod ( unroll ) ^[84]^[85] = "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)^[86]
transformations ^[87]

c-plane
inverted c plane = 1/c plane
unrolled

lambda plane

Points

Band-merging

the band-merging points are Misiurewicz points^[88]

Biaccessible

If there exist two distinct external rays landing at point we say that it is a biaccessible point. ^[89]

Center

Nucleus or center of hyperbolic component

A center of a hyperbolic component H is a parameter $c_{0}\in H\,$ ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." ^[90]

Synonyms :

Nucleus of a Mu-Atom ^[91]

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." ^[92]

Critical

A critical point^[93] of $f_{c}\,$ is a point $z_{cr}\,$ in the dynamical plane such that the derivative vanishes:

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

Since

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

implies

z_{cr}=0\,

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

$z_{0}$ is an initial point for Mandelbrot set iteration.^[94]

Cut

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).^[95] 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. ^[96] Cut rays can be used to construct puzzles.

Cut angle is an angle of cut ray.

fixed

Periodic point when period = 1

Feigenbaum

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

\delta

The Feigenbaum Point^[97] 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)^[98]

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

Generalized Feigenbaum points are :

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

Examples :

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

The Mandelbrot set is conjectured to be self- similar around generalized Feigenbaum points^[99] when the magnification increases by 4.6692 (the Feigenbaum Constant) and period is doubled each time^[100]

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

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

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

" a "Feigenbaum point" (an infinitely renormalizable parameter of bounded type, such as the famous Feigenbaum value which is the limit of the period-2 cascade of bifurcations), then Milnor's hairiness conjecture, proved by Lyubich, states that rescalings of the Mandelbrot set converge to the entire complex plane. So there is certainly a lot of thickness near such a point, although again this may not be what you are looking for. It may also prove computationally intensive to produce accurate pictures near such points, because the usual algorithms will end up doing the maximum number of iterations for almost all points in the picture." Lasse Rempe-Gillen^[101]

infinity

The point at infinity ^[102]" 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^[103]

Misiurewicz

Misiurewicz point^[104] = " 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 )^[105]
the branch points
tips in the Mandelbrot set ( tips of the midgets ) ^[106]

Characteristic Misiurewicz pointof the chaotic band of the Mandelbrot set is :^[107]

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-Feigenbaum

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

Parabolic point

parabolic points : this occurs when two singular points coallesce in a double singular point (parabolic point)^[108]

Periodic

Point z has period p under f if :

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

Pinching

"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 ^[109]

post-critical

A post-critical point is a point

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

where $z_{cr}$ is a critical point. ^[110]

precritical

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

root

The root point :

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

singular

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) / (z − a)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 ${\sqrt {z}}$ or $\log(z)$ 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 $z=0$ and $z=\infty$ for $\log(z)$ ) which are fixed in place.

Portrait

orbit portrait

types

There are two types of orbit portraits: primitive and satellite. ^[111]If $v$ is the valence of an orbit portrait ${\mathcal {P}}$ and $r$ is the recurrent ray period, then these two types may be characterized as follows:

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

Processes and phenomenona

Contraction and dilatation

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

Implosion and explosion

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)^[112]

Explosion is a :

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

Example : in exponential dynamics when λ> 1/e , the Julia set of $E_{\lambda }(z)=\lambda e^{z}$ is the entire plane.^[113]

Tuning

definition^[114]
examples
- subwake
- island

Radius

Conformal radius

Conformal radius of Siegel Disk ^[115]^[116]

Escape radius ( ER)

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

Minimal Escape Radius should be grater or equal to 2 :

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

Better estimation is :^[117]^[118]

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

Inner radius

Inner radius of Siegel Disc

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

Internal radius

Internal radius is a:

absolute value of multiplier $r=|\lambda |$

Sequences

A sequence is an ordered list of objects (or events).^[119]

A series is the sum of the terms of a sequence of numbers.^[120] Some times these names are not used as in above definitions.

Orbit

Orbit can be:

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

Set

Continuum

definition^[121]

Component

Components of parameter plane

mu-atom , ball, bud, bulb, decoration, lake and lakelet.^[122]

Islands

Names :

mini Mandelbrot set
'baby'-Mandelbrot set
island mu-molecules = embedded copy of the Mandelbrot Set^[123]
Bug
Island
Mandelbrotie
Midget

List of islands :

http://mrob.com/pub/mu-data/largest-islands.txt
http://mrob.com/pub/muency/largestislands.html
http://www.math.cornell.edu/~rperez/Documents/maximals.pdf
http://fraktal.republika.pl/mset_external_ray_mini.html
http://mathr.co.uk/mandelbrot/feature-database.csv.bz2 ( a database of all islands up to period 16, found by tracing external rays): period, islandhood, angled internal address, lower external angle numerator, denominator, upper numerator, denominator, orientation, size, centre realpart, imagpart

Primary and satellite

"Hyperbolic components come in two kinds, primitive and satellite, depending on the local properties of their roots." ^[124]

Child (Descendant ) and the parent

def ^[125]

Hyperbolic component of Mandelbrot set

Boundaries of hyperbolic components of Mandelbrot set

Domain is an open connected subset of a complex plane.

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

A center of a H is a parameter $c_{0}\in H\,$ ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." ^[126]

A hyperbolic component is narrow if it contains no component of equal or lesser period in its wake ^[127]

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

Limb

13/34 limb and wake on the left image

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

Wake

Wakes of Mandelbrot Set to Period 10

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

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

Components of dynamical plane

Components of interior of Filled Julia set
Inverse iteration of Siegel disc component

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

Domain

Domain in mathematical analysis it is an open connected set

Jordan domain

"A Jordan domain^[128] 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." ^[129]

Dwell bands

"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). " ^[130] Other names:

level sets of integer escape time

Flower

Lea-Fatu flower

Invariant

"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" ^[131]

Examples :

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

Level set

a level set of a real-valued function f^[132] ( see also dwell band)
Level set methods (LSM)

Planar set

a non-separating planar set is a set whose complement in the plane is connected.^[133]

Sepal

Target set

How target set is changing along internal ray 0

Elliptic case

Target set in elliptic case = inner circle

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

Hyperbolic case

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

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

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

Exterior of circle

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

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

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

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

Exterior of square

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

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

Parabolic case : petal

trap in parabolic case

In the parabolic case target set shoul be iside petal

Trap

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 ).

Surgery

surgery in differential topology ^[134]

Surgery on the circle
Surgery on the sphere

Links:

http://mathoverflow.net/questions/tagged/surgery-theory

Test

Bailout test or escaping test

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.^[135] It allows to find 2 sets :

escaping points ( it should be also the whole basing of attraction to infinity)^[136]
not escaping points ( it should be the complement of basing of attraction to infinity)

In practice for given IterationMax and Escape Radius :

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

If $z_{n}$ is in the target set $T\,$ then $z_{0}$ is escaping to infinity ( bailouts ) after n forward iterations ( steps).^[137]

The output of test can be :

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

Tree

Farey tree

Farey tree = Farey sequence as a tree

Hubbard tree

"Hubbard trees are finite planar trees, equipped with self-maps, which classify postcritically finite polynomials as holomorphic dynamical systems on the complex plane." ^[138]

References

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