Definitions

# Angle

## Types

Compare different types of angles :

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 :

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# Curves

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## Circle

### Unit circle

Unit circle $\partial D\,$ is a boundary of unit disk[1]

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

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

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

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## 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[2]

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## 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[3][4]

It is a model of Mandelbrot or Julia set.

A lamination, L, is a union of leaves and the unit circle which satisfies :[5]

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

Chords = leaves = arcs

A leaf on the unit disc is a path connecting two points on the unit circle. [6]

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## 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 [7]( extended rays = external + internal ray) in the complex plane connecting each of the post-critical points to infnity [8] See spider algorithm for more.

# Derivative

Derivative of Iterated function (map)

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## 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 :

$z_0 = 0 \qquad\qquad z'_0 = 1 \,$
$z_1 = c \qquad\qquad z'_1 = 1 \,$
$z_2 = c^2+c \qquad z'_2 = 2c+1 \,$

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## 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

# Magnitude

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

# Multiplier

Multiplier of periodic z-point : [9]

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[10]
• an evolution function[11] of the discrete nonlinear dynamical system[12]
$z_{n+1} = f_c(z_n) \,$

is called map $f_c$ :

$f_c : z \to z^2 + c. \,$

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### Forms

#### c form : $z^2+c$

• 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


• 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 $\theta = \frac {p}{q}$

Multiplier of fixed point :

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

When one wants change from lambda to c :[14]

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

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[15]

double InternalAngleInTurns;
double t = InternalAngleInTurns *2*M_PI; // from turns to radians
double Cx, Cy; /* C = Cx+Cy*i */
// main cardioid



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
*/
ToCircle(t,r):=r*%e^(%i*t*2*%pi);

(
[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;

/* --------- main -------------- */
for Denominator:1 thru DenominatorMax step 1 do
(
InternalAngle: Numerator/Denominator,
display(Denominator),
display(c),
/* compute fixed point */
alfa:float(rectform((1-sqrt(1-4*c))/2)), /* alfa fixed point */
display(alfa)
)$ ↑Jump back a section ## Doubling map definition [16] • 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

-- 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;
}

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## 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 "[18]

"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)

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## 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.[19]

# Orbit

Critical orbit[20] is a forward orbit of critical point[21][22]

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.[23]

More is here

# Plane

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## Dynamic plane

• z-plane for fc(z)= z^2 + c
• z-plane for fm(z)= z^2 + m*z
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See :[24]

# Points

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## Biaccessible point

If there exist two distinct external rays landing at point we say that it is a biaccessible point. [30]

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## 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." [31]

### 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." [32]

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## Critical point

Critical point [33]

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## 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).[34] 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. [35] Cut rays can be used to construct puzzles.

Cut angle is an angle of cut ray.

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## Periodic point

Point z has period p under f if :

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

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## 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 [36]
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## A post-critical point

A post-critical point is a point

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

where $z_{cr}$ is a critical point. [37]

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Conformal radius of Siegel Disk [38][39]

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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 :[40][41]

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

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

# Set

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## Continuum

definition[42]

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## 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." [43]

A hyperbolic component is narrow if it contains no component of equal or lesser period in its wake [44]

#### Wake

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

### Components of dynamical plane

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

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## Domain

Domain in mathematical analysis it is an open connected set

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## Planar set

a non-separating planar set is a set whose complement in the plane is connected.[45]

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## 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 :

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

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

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## 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

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## 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)[46]
• 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).[47]

The output of test can be :

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

# References

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