Fractals/Iterations in the complex plane/def cqp

Definitions

Angle

Types

Compare different types of angles :

• external ( point of set's exterior )
• internal ( point of component's interior )
• plain ( argument of complex number )

	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

Units

Curves

Circle

Inner circle

Unit circle

Unit circle $\partial D\,$ is a boundary of unit disk

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

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

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

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

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

It is a model of Mandelbrot or Julia set.

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

• 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.

Ray

External ray

Internal ray

Dynamic internal and external 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 ( extended rays = external + internal ray) in the complex plane connecting each of the post-critical points to infnity See spider algorithm for more.

Derivative

Derivative of Iterated function (map)

Derivative with respect to c

On parameter plane :

• $c$ is a variable
• $z_0 = 0$ is constant

$\frac{d}{dc} f^{(p)} _c (z_0) = z'_p \,$

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 :

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

Iteration

Iteration

Magnitude

magnitude of the point = it's distance from the origin

Multiplier

Multiplier of periodic z-point :

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	$z^8 + 4cz^6 + 6c^2z^4 + 2cz^4 + 4c^3z^2 + 4c^2z^2 + c^4 + 2c^3 + c^2 + c$	$8z^7 + 24cz^5 + 24c^2z^3 + 8cz^3 + 8c^3z + 8c^2z$

It is used to :

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

Map

• Iterated function = map
• an evolution function of the discrete nonlinear dynamical system

$z_{n+1} = f_c(z_n) \,$

is called map $f_c$ :

$f_c : z \to z^2 + c. \,$

Complex quadratic map

Forms

c form : $z^2+c$

quadratic map

• 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

$\ f^{(0)} _c (z) = z = z_0$

$\ f^{(1)} _c (z) = f_c(z) = z_1$

...

$\ f^{(p)} _c (z) = f_c(f^{(p-1)} _c (z))$

or with subscripts :

$\ z_p = f^{(p)} _c (z_0)$

• 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 : $z^2+\lambda z$

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 :

$\lambda = r e^{2 \pi \theta i}$

When one wants change from lambda to c :

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

or from c to lambda :

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

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

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)
 )$

Doubling map

definition

• 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

-- 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;
}

First return 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 "

"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)

Multiplier map

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

$\lambda : \Eta \to \mathbb{D}$

Multiplier map is a conformal isomorphism.

Orbit

Critical orbit is a forward orbit of critical point

Here are images of critical orbits

Period

The smallest positive integer value k for which this equality

$f^k(z_0) = z_0$

holds is the period of the orbit.

More is here

Plane

Dynamic plane

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

Parameter plane

See :

• exponential plane ( map)
• flatten' the cardiod ( unroll )
• transformations

• 

  c-plane

• 

  inverted c plane = 1/c plane

• 

  lambda plane

Points

Biaccessible point

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

Center

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."

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."

Critical point

Critical point

Cut point, ray and angle

The "neck" of this eight-like figure is a cut-point.

Cut point k of set S is a point for which set S-k is dissconected ( consist of 2 or more sets). 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. Cut rays can be used to construct puzzles.

Cut angle is an angle of cut ray.

Periodic point

Point z has period p under f if :

$z : \ f^{p} (z) = z$

Pinching points

"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

A post-critical point

A post-critical point is a point

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

where $z_{cr}$ is a critical point.

Radius

Conformal radius

Conformal radius of Siegel Disk

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 :

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

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 $r = |\lambda|$

Set

Continuum

definition

Component

Components of parameter plane

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."

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

Wake

Wake is the region of parameter plane enclosed by its two external rays

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

Planar set

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

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

In the parabolic case target set is a 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 ).

Test

Bailout 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. It allows to find 2 sets :

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

The output of test can be :

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

References