Fractals/Iterations in the complex plane/def cqp
Definitions
Contents
 1 Address
 2 Angle
 3 Coordinate
 4 Curves
 5 Derivative
 6 Iteration
 7 Magnitude
 8 Multiplier
 9 Map
 10 Number
 11 Orbit
 12 Parameter
 13 Period
 14 Plane
 15 Points
 16 Polynomial
 17 Portrait
 18 Processes
 19 Radius
 20 Sequences
 21 Set
 22 Test
 23 References
Address[edit]
Internal[edit]
Internal addresses describe the combinatorial structure of the Mandelbrot set.
angled[edit]
Angled internal address is an extension of internal address
Angle[edit]
Types of angle[edit]
external angle  internal angle  plain angle  

parameter plane  
dynamic plane 
where :
 is a multiplier map
 is a Boettcher function
external[edit]
The external angle is a angle of point of set's exterior. It is the same on all points on the external ray
internal[edit]
The internal angle 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
plain[edit]
The plain angle is an agle of complex point = it's argument ^{[1]}
Units[edit]
 turns
 degrees
 radians
Number types[edit]
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 :^{[2]}
where :
Coordinate[edit]
 Fatou coordinate for every repelling and attracting petal ( linearization of function near parabolic fixed point )
Curves[edit]
Circle[edit]
Inner circle[edit]
Unit circle[edit]
Unit circle is a boundary of unit disk^{[3]}
where coordinates of point of unit circle in exponential form are :
Critical curves[edit]
Diagrams of critical polynomials are called critical curves.^{[4]}
These curves create skeleton of bifurcation diagram.^{[5]} (the dark lines^{[6]})
Isocurves[edit]
Equipotential lines[edit]
Jordan curve[edit]
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^{[7]}
Lamination[edit]
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^{[8]}^{[9]}
It is a model of Mandelbrot or Julia set.
A lamination, L, is a union of leaves and the unit circle which satisfies :^{[10]}
 leaves do not cross (although they may share endpoints) and
 L is a closed set.
Leaf[edit]
Chords = leaves = arcs
A leaf on the unit disc is a path connecting two points on the unit circle. ^{[11]}
Ray[edit]
External ray[edit]
Internal ray[edit]
Internal rays are :
 dynamic ( on dynamic plane , inside filled Julia set )
 parameter ( on parameter plane , inside Mandelbrot set )
Spider[edit]
A spider S is a collection of disjoint simple curves called legs ^{[12]}( extended rays = external + internal ray) in the complex plane connecting each of the postcritical points to infnity ^{[13]} See spider algorithm for more.
Derivative[edit]
Derivative of Iterated function (map)
Derivative with respect to c[edit]
On parameter plane :
 is a variable
 is constant
This derivative can be found by iteration starting with
and then
This can be verified by using the chain rule for the derivative.
 Maxima CAS function :
dcfn(p, z, c) := if p=0 then 1 else 2*fn(p1,z,c)*dcfn(p1, z, c)+1;
Example values :
Derivative with respect to z[edit]
is first derivative with respect to c.
This derivative can be found by iteration starting with
and then :
Iteration[edit]
Magnitude[edit]
magnitude of the point = it's distance from the origin
Multiplier[edit]
Multiplier of periodic zpoint : ^{[14]}
Math notation :
Maxima CAS function for computing multiplier of periodic cycle :
m(p):=diff(fn(p,z,c),z,1);
where p is a period. It takes period as an input, not z point.
period  

1  
2  
3 
It is used to :
 compute stability index of periodic orbit ( periodic point) = ( where r is a n internal radius
 multiplier map
Map[edit]
 Iterated function = map^{[15]}
 an evolution function^{[16]} of the discrete nonlinear dynamical system^{[17]}
is called map :
Complex quadratic map[edit]
Forms[edit]
c form : [edit]
quadratic map^{[18]}
 math notation :
 Maxima CAS function :
f(z,c):=z*z+c;
(%i1) z:zx+zy*%i; (%o1) %i*zy+zx (%i2) c:cx+cy*%i; (%o2) %i*cy+cx (%i3) f:z^2+c; (%o3) (%i*zy+zx)^2+%i*cy+cx (%i4) realpart(f); (%o4) zy^2+zx^2+cx (%i5) imagpart(f); (%o5) 2*zx*zy+cy
Iterated quadratic map
 math notation
...
or with subscripts :
 Maxima CAS function :
fn(p, z, c) := if p=0 then z elseif p=1 then f(z,c) else f(fn(p1, z, c),c);
zp:fn(p, z, c);
lambda form : [edit]
More description Maxima CAS code ( here m not lambda is used ) :
(%i2) z:zx+zy*%i; (%o2) %i*zy+zx (%i3) m:mx+my*%i; (%o3) %i*my+mx (%i4) f:m*z+z^2; (%o4) (%i*zy+zx)^2+(%i*my+mx)*(%i*zy+zx) (%i5) realpart(f); (%o5) zy^2my*zy+zx^2+mx*zx (%i6) imagpart(f); (%o6) 2*zx*zy+mx*zy+my*zx
Switching between forms[edit]
Start from :
 internal angle
 internal radius r
Multiplier of fixed point :
When one wants change from lambda to c :^{[19]}
or from c to lambda :
Example values :
r  c  fixed point alfa  fixed point  

1/1  1.0  0.25  0.5  1.0  0 
1/2  1.0  0.75  0.5  1.0  0 
1/3  1.0  0.64951905283833*i0.125  0.43301270189222*i0.25  0.86602540378444*i0.5  0 
1/4  1.0  0.5*i+0.25  0.5*i  i  0 
1/5  1.0  0.32858194507446*i+0.35676274578121  0.47552825814758*i+0.15450849718747  0.95105651629515*i+0.30901699437495  0 
1/6  1.0  0.21650635094611*i+0.375  0.43301270189222*i+0.25  0.86602540378444*i+0.5  0 
1/7  1.0  0.14718376318856*i+0.36737513441845  0.39091574123401*i+0.31174490092937  0.78183148246803*i+0.62348980185873  0 
1/8  1.0  0.10355339059327*i+0.35355339059327  0.35355339059327*i+0.35355339059327  0.70710678118655*i+0.70710678118655  0 
1/9  1.0  0.075191866590218*i+0.33961017714276  0.32139380484327*i+0.38302222155949  0.64278760968654*i+0.76604444311898  0 
1/10  1.0  0.056128497072448*i+0.32725424859374  0.29389262614624*i+0.40450849718747  0.58778525229247*i+0.80901699437495 
One can easily compute parameter c as a point c inside main cardioid of Mandelbrot set :
of period 1 hyperbolic component ( main cardioid) for given internal angle ( rotation number) t using this c / cpp code by Wolf Jung^{[20]}
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/2w*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((1sqrt(14*c))/2)), /* alfa fixed point */ display(alfa) )$
Doubling map[edit]
definition ^{[21]}
 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 doublingmap (ratioangle) " 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 ratioangle)) (d (denominator ratioangle))) (setq n (mod (* n 2) d)) ; (2 * n) modulo d (/ n d))) ; result = n/d
 Haskell function^{[22]}
 by Claude HeilandAllen  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[edit]
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." ^{[23]}. Inverse of doubling map is multivalued function.
In Maxima CAS :
InvDoublingMap(r):= [r/2, (r+1)/2];
Note that difference between these 2 preimages
is half a turn = 180 degrees = Pi radians.
First return map[edit]
definition ^{[24]}
"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 "^{[25]}
"If x is a periodic point of period p for f and U is a neighborhood of x, the composition maps U to another neighborhood V of x. This locally defined map is the return map for x." ( W P Thurston : On the geometry and dynamics of Iterated rational maps)
Multiplier map[edit]
Multiplier map gives an explicit uniformization of hyperbolic component by the unit disk :
Multiplier map is a conformal isomorphism.^{[26]}
Number[edit]
Rotation number[edit]
The rotation number^{[27]}^{[28]}^{[29]} 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 ^{[30]}
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/qlimb, 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
Orbit[edit]
Backward[edit]
Critical[edit]
Forward orbit^{[31]} of a critical point^{[32]}^{[33]} is called a critical orbit. Critical orbits are very important because every attracting periodic orbit^{[34]} attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.^{[35]}^{[36]}
This orbit falls into an attracting periodic cycle.
Here are images of critical orbits
Forward[edit]
Inverse[edit]
Inverse = Backward
Parameter[edit]
Parameter ( point of parameter plane ) " is renormalizable if restriction of some of its iterate gives a polinomiallike map of the same or lower degree. " ^{[37]}
Period[edit]
The smallest positive integer value p for which this equality
holds is the period of the orbit.^{[38]}
is a point of periodic orbit ( limit cycle ) .
More is here
Plane[edit]
Planes ^{[39]}
Dynamic plane[edit]
 zplane for fc(z)= z^2 + c
 zplane for fm(z)= z^2 + m*z
Parameter plane[edit]
See :^{[40]}
Types of the parameter plane :
 cplane ( standard plane )
 exponential plane ( map) ^{[41]}^{[42]}
 flatten' the cardiod ( unroll ) ^{[43]}^{[44]} = "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 204205 of The Science Of Fractal Images)^{[45]}
 transformations ^{[46]}
Points[edit]
Bandmerging points[edit]
the bandmerging points are Misiurewicz points^{[47]}
Biaccessible point[edit]
If there exist two distinct external rays landing at point we say that it is a biaccessible point. ^{[48]}
Center[edit]
Center of hyperbolic component[edit]
A center of a hyperbolic component H is a parameter ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." ^{[49]}
Center of Siegel Disc[edit]
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." ^{[50]}
Critical point[edit]
A critical point^{[51]} of is a point in the dynamical plane such that the derivative vanishes:
Since
implies
we see that the only (finite) critical point of is the point .
is an initial point for Mandelbrot set iteration.^{[52]}
Cut point, ray and angle[edit]
Cut point k of set S is a point for which set Sk is dissconected ( consist of 2 or more sets).^{[53]} 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. ^{[54]} Cut rays can be used to construct puzzles.
Cut angle is an angle of cut ray.
Feigenbaum Point[edit]
The Feigenbaum Point^{[55]} 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)^{[56]}
Generalized Feigenbaum points are :
 the limit of the periodq cascade of bifurcations
 landing points of parameter ray or rays with irrational angles
Examples :
 .1528+1.0397i)
The Mandelbrot set is conjectured to be self similar around generalized Feigenbaum points^{[57]} when the magnification increases by 4.6692 (the Feigenbaum Constant) and period is doubled each time^{[58]}

n Period = 2^n Bifurcation parameter = c_{n} Ratio 1 2 0.75 N/A 2 4 1.25 N/A 3 8 1.3680989 4.2337 4 16 1.3940462 4.5515 5 32 1.3996312 4.6458 6 64 1.4008287 4.6639 7 128 1.4010853 4.6682 8 256 1.4011402 4.6689 9 512 1.401151982029 10 1024 1.401154502237 infinity 1.4011551890 ...
Bifurcation parameter is a root point of period = 2^n component. This series converges to the Feigenbaum point c = −1.401155
The ratio in the last column converges to the first Feigenbaum constant.
Misiurewicz point[edit]
Misiurewicz point^{[59]}
Characteristic Misiurewicz pointof the chaotic band of the Mandelbrot set is :^{[60]}
 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
the MyrbergFeigenbaum point[edit]
MF = the MyrbergFeigenbaum point is the different name for the Feigenbaum Point.
Periodic point[edit]
Point z has period p under f if :
Pinching points[edit]
"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 welldefined 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 ^{[61]}
A postcritical point[edit]
A postcritical point is a point
where is a critical point. ^{[62]}
root point[edit]
The root point :
 has a rotational number 0
 it is a biaccesible point ( landing point of 2 external rays )
Polynomial[edit]
Critical polynomial[edit]
so
These polynomials are used for finding :
 centers of period n Mandelbrot set components. Centers are roots of nth critical polynomials ( points where critical curve Qn croses x axis )
 Misiurewicz points
Portrait[edit]
orbit portrait[edit]
types[edit]
There are two types of orbit portraits: primitive and satellite. ^{[63]}If is the valence of an orbit portrait and is the recurrent ray period, then these two types may be characterized as follows:
 Primitive orbit portraits have and . Every ray in the portrait is mapped to itself by . Each is a pair of angles, each in a distinct orbit of the doubling map. In this case, is the base point of a baby Mandelbrot set in parameter space.
 Satellite ( nonprimitive ) orbit portraits have . In this case, all of the angles make up a single orbit under the doubling map. Additionally, is the base point of a parabolic bifurcation in parameter space.
Processes[edit]
Contraction and dilatation[edit]
 the contraction z → z/2
 the dilatation z → 2z.
Implosion and explosion[edit]
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)
Explosion is a :
 is a sudden change of quality fuatures of the object in an extreme manner,
 the opposite of implosion
Example : in exponential dynamics when λ> 1/e , the Julia set of is the entire plane.^{[64]}
Radius[edit]
Conformal radius[edit]
Conformal radius of Siegel Disk ^{[65]}^{[66]}
Escape radius ( ER)[edit]
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 :
Better estimation is :^{[67]}^{[68]}
Inner radius[edit]
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[edit]
Internal radius is a:
 absolute value of multiplier
Sequences[edit]
A sequence is an ordered list of objects (or events).^{[69]}
A series is the sum of the terms of a sequence of numbers.^{[70]} Some times these names are not used as in above definitions.
The upper principal sequence of rotational number around the main cardioid of Mandelbrot set^{[71]}
n  rotation number = 1/n  parameter c 

2  1/2  0.75 
3  1/3  0.64951905283833*i0.125 
4  1/4  0.5*i+0.25 
5  1/5  0.32858194507446*i+0.35676274578121 
6  1/6  0.21650635094611*i+0.375 
7  1/7  0.14718376318856*i+0.36737513441845 
8  1/8  0.10355339059327*i+0.35355339059327 
9  1/9  0.075191866590218*i+0.33961017714276 
10  1/10  0.056128497072448*i+0.32725424859374 
Set[edit]
Continuum[edit]
definition^{[72]}
Component[edit]
Components of parameter plane[edit]
Hyperbolic component of Mandelbrot set[edit]
Domain is an open connected subset of a complex plane.
"A hyperbolic component H of Mandelbrot set is a maximal domain (of parameter plane) on which has an attracting periodic orbit.
A center of a H is a parameter ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." ^{[73]}
A hyperbolic component is narrow if it contains no component of equal or lesser period in its wake ^{[74]}
Limb[edit]
p/q limb is a part of Mandelbrot set contained inside p/q wake
Wake[edit]
Wake is the region of parameter plane enclosed by its two external rays landing on the same root point.
Components of dynamical plane[edit]
In case of Siegel disc critical orbit is a boundary of component containing Siegel Disc.
Domain[edit]
Domain in mathematical analysis it is an open connected set
Planar set[edit]
a nonseparating planar set is a set whose complement in the plane is connected.^{[75]}
Target set[edit]
Elliptic case[edit]
For the elliptic dynamics, when there is a Siegel disc, the target set is an inner circle
Hyperbolic case[edit]
Infinity is allways hyperbolic attractor for forward iteration of polynomials. Target set here is an exterior of any shape containing all point of Julia set ( and it's interior). There are also other hyperbolic attractors.
In case of forward iteration target set is an arbitrary set on dynamical plane containing infinity and not containing points of filled Julia set.
For escape time algorithms target set determines the shape of level sets and curves. It does not do it for other methods.
Exterior of circle[edit]
This is typical target set. It is exterior of circle with center at origin and radius =ER :
Radius is named escape radius ( ER ) or bailout value.
Circle of radius=ER centered at the origin is :
Exterior of square[edit]
Here target set is exterior of square of side length centered at origin
Parabolic case[edit]
In the parabolic case target set is a petal
Trap[edit]
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[edit]
Bailout test[edit]
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)^{[76]}
 not escaping points ( it should be the complement of basing of attraction to infinity)
In practice for given IterationMax and Escape Radius :
 some pixels from set of not escaping points may contain points that escape after more iterations then IterationMax ( increase IterMax )
 some pixels from escaping set may contain points from thin filaments not choosed by maping from integer to world ( use DEM )
If is in the target set then is escaping to infinity ( bailouts ) after n forward iterations ( steps).^{[77]}
The output of test can be :
 boolean ( yes/no)
 integer : integer number (value of the last iteration)
References[edit]
 ↑ argument of complex number
 ↑ A Method to Solve the Limitations in Drawing External Rays of the Mandelbrot Set M. Romera, G. Pastor, A. B. Orue, A. Martin, M.F. Danca, and F. Montoya
 ↑ Unit circle in wikipedia
 ↑ The Road to Chaos is Filled with Polynomial Curves by Richard D. Neidinger and R. John Annen III. American Mathematical Monthly, Vol. 103, No. 8, October 1996, pp. 640653
 ↑ Hao, Bailin (1989). Elementary Symbolic Dynamics and Chaos in Dissipative Systems. World Scientific. ISBN 9971506823. http://power.itp.ac.cn/~hao/.
 ↑ M. Romera, G. Pastor and F. Montoya, "Misiurewicz points in onedimensional quadratic maps", Physica A, 232 (1996), 517535. Preprint
 ↑ wikipedia : Jordan curve theorem
 ↑ Modeling Julia Sets with Laminations: An Alternative Definition by Debra Mimbs
 ↑ Laminations of the unit disk with irrational rotation gaps by John C. Mayer
 ↑ Rational maps represented by both rabbit and aeroplane matings Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Freddie R. Exall July 2010
 ↑ Rational maps represented by both rabbit and aeroplane matings Thesis submitted in accordance with the requirements of the University of Liverpool for the degree of Doctor in Philosophy by Freddie R. Exall July 2010
 ↑ Iterated Monodromy Groups of Quadratic Polynomials, I Laurent Bartholdi, Volodymyr V. Nekrashevych
 ↑ GROWTH OF GROUPS DEFINED BY AUTOMATA : ASHLEY S. DOUGHERTY, LYDIA R. KINDELIN, AARON M. REAVES, ANDREW J. WALKER, AND NATHANIEL F. ZAKAHI
 ↑ Multiplier at wikipedia
 ↑ Iterated function (map) in wikipedia
 ↑ evolution function
 ↑ the discrete nonlinear dynamical system
 ↑ Complex quadratic map in wikipedia
 ↑ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
 ↑ Mandel: software for real and complex dynamics by Wolf Jung
 ↑ wikipedia : Dyadic transformation
 ↑ lavaurs' algorithm in Haskell with SVG output by Claude HeilandAllen
 ↑ SYMBOLIC DYNAMICS AND SELFSIMILAR GROUPS by VOLODYMYR NEKRASHEVYCH
 ↑ Poincaré map
 ↑ General principles of chaotic dynamics by P.B. Persson , C.D. Wagner
 ↑ Conformal Geometry and Dynamics of Quadratic Polynomials Mikhail Lyubich
 ↑ wikipedia : Rotation number
 ↑ scholarpedia : Rotation_theory
 ↑ The Fractal Geometry of the Mandelbrot Set II. How to Count and How to Add Robert L. Devaney
 ↑ Complex systems simulation Curso 20122013 by Antonio Giraldo and María Asunción Sastre
 ↑ wikipedia : orbit (dynamics)
 ↑ Wikipedia : Complex quadratic polynomial  Critical point
 ↑ MandelOrbits  A visual realtime trace of Mandelbrot iterations by Ivan Freyman
 ↑ wikipedia : Periodic points of complex quadratic mappings
 ↑ M. Romera, G. Pastor, and F. Montoya : Multifurcations in nonhyperbolic fixed points of the Mandelbrot map. Fractalia 6, No. 21, 1012 (1997)
 ↑ Burns A M : Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set. Mathematics Magazine, Vol. 75, No. 2 (Apr., 2002), pp. 104116
 ↑ Ouadraticlike maps and Renormalization by Nuria Fagella
 ↑ scholarpedia : Periodic Orbit for a Map
 ↑ wikipedia : Complex_quadratic_polynomial  Planes
 ↑ Alternate Parameter Planes by David E. Joyce
 ↑ muency : exponential map by R Munafo
 ↑ Exponential mapping and OpenMP by Claude HeilandAllen
 ↑ Linas Vepstas : Self Similar?
 ↑ the flattened cardioid of a Mandelbrot by Tom Rathborne
 ↑ Stretching cusps by Claude HeilandAllen
 ↑ Twisted Mandelbrot Sets by Eric C. Hill
 ↑ doubling bifurcations on complex plane by E Demidov
 ↑ On biaccessible points in the Julia set of the family z(a+z^{d}) by Mitsuhiko Imada
 ↑ Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper
 ↑ Siegel disks by Xavier Buff and Arnaud Ch ́ritat e Univ. Toulouse Roma, April 2009
 ↑ wikipedia : Critical point (mathematics)
 ↑ Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations
 ↑ Cut point in wikipedia
 ↑ On local connectivity for the Julia set of rational maps : Newton’s famous example By P. Roesch
 ↑ muency : feigenbaum point
 ↑ On Periodic and Chaotic Regions in the Mandelbrot Set by G. Pastor , M. Romera, G. Álvarez, D. Arroyo and F. Montoya
 ↑ fractalfaq : section 6
 ↑ Period doubling and Feigenbaum's scaling be E Demidov
 ↑ wikipedia : Misiurewicz point
 ↑ G. Pastor, M. Romera, G. Álvarez, D. Arroyo and F. Montoya, "On periodic and chaotic regions in the Mandelbrot set", Chaos, Solitons & Fractals, 32 (2007) 1525
 ↑ http://www.mndynamics.com/indexp.html%7C program Mandel by Wolf Jung , demo 2 page 3
 ↑ GROWTH OF GROUPS DEFINED BY AUTOMATA : ASHLEY S. DOUGHERTY, LYDIA R. KINDELIN, AARON M. REAVES, ANDREW J. WALKER, AND NATHANIEL F. ZAKAHI
 ↑ wikipedia : Orbit portrait
 ↑ CANTOR BOUQUETS, EXPLOSIONS, AND KNASTER CONTINUA: DYNAMICS OF COMPLEX EXPONENTIALS by Robert L. Devaney Publicacions Matematiques, Vol 43 (1999), 27–54.
 ↑ wikipedia : Conformal radius
 ↑ scholarpedia : Quadratic Siegel disks
 ↑ Julia Sets of Complex Polynomials and Their Implementation on the Computer by Christoph Martin Stroh
 ↑ fractalforums: bounding circle of julia sets by knighty
 ↑ wikipedia : Sequence
 ↑ wikipedia : series
 ↑ Mandel Set Combinatorics : Principal Series
 ↑ wikipedia : Continuum in set theory
 ↑ Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper
 ↑ Internal addresses in the Mandelbrot set and irreducibility of polynomials by Dierk Schleicher
 ↑ A. Blokh, X. Buff, A. Cheritat, L. Oversteegen The solar Julia sets of basic quadratic Cremer polynomials, Ergodic Theory and Dynamical Systems , 30 (2010), #1, 5165
 ↑ wikipedia : Escaping set
 ↑ fractint doc : bailout