Fractals/Iterations in the complex plane/def cqp
Definitions
Contents
 1 Address
 2 Angle
 3 Bifurcation
 4 Coordinate
 5 Curves
 6 Density
 7 Derivative
 8 Discretization
 9 Discriminant
 10 Dynamics
 11 equation
 12 Function
 13 glitches
 14 graf
 15 Invariants
 16 Interval
 17 Iteration
 18 Itinerary
 19 Magnitude
 20 Map
 21 Multiplier
 22 Normalize
 23 Number
 24 Orbit
 25 Parameter
 26 Paritition
 27 Period
 28 Perturbation
 29 Plane
 30 Points
 30.1 Bandmerging
 30.2 Biaccessible
 30.3 Center
 30.4 Critical
 30.5 Cut
 30.6 fixed
 30.7 Feigenbaum
 30.8 Fibonacci
 30.9 infinity
 30.10 Misiurewicz
 30.11 MyrbergFeigenbaum
 30.12 Parabolic point
 30.13 Periodic
 30.14 Pinching
 30.15 postcritical
 30.16 precritical
 30.17 reference point
 30.18 root
 30.19 singular
 31 Portrait
 32 Precision
 33 Processes or transformations and phenomenona
 34 Aliasing and antialising
 35 Radius
 36 Sequences
 37 Set
 38 smooth
 39 Surgery
 40 Test
 41 References
Address[edit]
Internal[edit]
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[edit]
Angled internal address is an extension of internal address
This address describes period 6 component which is a satelite of period 3 component.
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^{[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
 "The internal angles start at 0, at the cusp, and increase counterclockwise. " Robert Munafo^{[4]}
plain[edit]
The plain angle is an agle of complex point = it's argument ^{[5]}
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 :^{[6]}
where :
Bifurcation[edit]
 Numerical Bifurcation Analysis of Maps
 MatCont^{[7]}
Coordinate[edit]
 Fatou coordinate for every repelling and attracting petal ( linearization of function near parabolic fixed point )
 Boettcher
 Koenigs
Curves[edit]
Types:
 topology:
 closed versus open
 simple versus not simple
 other properities:
 invariant
 critical
Description^{[8]}
 plane curve = it lies in a plane.
 closed = it starts and ends at the same place.
 simple = it never crosses itself.
closed[edit]
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.^{[9]}
 complex Closed Curve ( not simple = nonsimple ) It divides the plane into more than two regions. Example : Lemniscates.
"nonselfintersecting continuous closed curve in plane" = "image of a continuous injective function from the circle to the plane"
Circle[edit]
Inner circle[edit]
Unit circle[edit]
Unit circle is a boundary of unit disk^{[10]}
where coordinates of point of unit circle in exponential form are :
Critical curves[edit]
Diagrams of critical polynomials are called critical curves.^{[11]}
These curves create skeleton of bifurcation diagram.^{[12]} (the dark lines^{[13]})
Escape lines[edit]
"If the escape radius is equal to 2 the contour lines have a contact point (c= 2) and cannot be considered as equipotential lines" ^{[14]}
Invariant[edit]
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^{[15]}^{[16]}^{[17]}
 curve is invariant for a system of ordinary differential equations^{[18]}
Isocurves[edit]
Equipotential lines[edit]
Equipotential lines = Isocurves of complex potential
"If the escape radius is greater than 2 the contour lines are equipotential lines" ^{[19]}
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^{[20]}
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^{[21]}^{[22]}
It is a model of Mandelbrot or Julia set.
A lamination, L, is a union of leaves and the unit circle which satisfies :^{[23]}
 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.^{[24]}
Open curve[edit]
Curve which is not closed. Examples : line, ray.
Ray[edit]
Rays are :
 invariant curves
 dynamic or parameter
 external or internal
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 ^{[25]}( extended rays = external + internal ray) in the complex plane connecting each of the postcritical points to infnity ^{[26]}
See :
Vein[edit]
"A vein in the Mandelbrot set is a continuous, injective arc inside in the Mandelbrot set"
"The principal vein is the vein joining to the main cardioid" (Entropy, dimension and combinatorial moduli for onedimensional dynamical systems. A dissertation by Giulio Tiozzo )
Density[edit]
density of the image[edit]
Dense image^{[27]}^{[28]}^{[29]}
 downsaling with gamma correction^{[30]}
 path finding^{[31]}
 supersampling: "ots of detail but fractal fades away as you get more accurate, as n increases in nxn supersampling" TGlad
Derivative[edit]
 of the function f with respect to variable ...
 following the derivative^{[32]}
Discretization[edit]
discretization^{[33]} and its reverse ^{[34]}
Discriminant[edit]
In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them.
Dynamics[edit]
 symbolic^{[35]}^{[36]}^{[37]}
 complex ^{[38]}
 Arithmetic
 combinatorial
symbolic[edit]
"Symbolic dynamics encodes :
 a dynamical system by a shift map on a space of sequences over finite alphabet using Markov paritition of the space
 the points of space by their itineraries with respect to the paritition " ( Volodymyr Nekrashevych  Symbolic dynamics and selfsimilar groups
equation[edit]
differential[edit]
differential equations
 exact analytic solutions.
 approximated solution
 use perturbation theory to approximate the solutions
Function[edit]
Derivative[edit]
Derivative of Iterated function (map)^{[39]}
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 ( compute derivative before next z because it uses previous values of z):
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 :
It can be used for:
 the distance estimation method for drawing a Mandelbrot set ( DEM/M )
Derivative with respect to z[edit]
is first derivative with respect to z.
This derivative can be found by iteration starting with
and then :
It can be used for :
 computing the external distance to the Julia set ( DEM/J )
 detection of interior points of Mandelbrot set componnets^{[40]}
The Schwarzian Derivative[edit]
The Schwarzian Derivative ^{[41]} ^{[42]}
Wirtinger derivatives[edit]
gradient[edit]
Germ[edit]
Germ ^{[43]} of the function f in the neighborhood of point z is a set of the functions g which are indistinguishable in that neighborhood
See :
map[edit]
 differences between map and the function ^{[44]}
 Iterated function = map^{[45]}
 an evolution function^{[46]} of the discrete nonlinear dynamical system^{[47]}
is called map :
types[edit]
Polynomial[edit]
Critical[edit]
Critical polynomial :
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
postcritically finite[edit]
a postcritically finite polynomial = all critical points have finite orbit
meromorphic[edit]
meromorphic maps: Those with NO FINITE, NONATTRACTING FIXED POINTS^{[48]}
Resurgent[edit]
"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^{[49]}
glitches[edit]
glitches
 Incorrect parts of renders^{[50]} using perturbation techique
 pixel which dynamics differ significantly from the dynamics of the reference pixel^{[51]}
How to detect glitches:
 heuristic developed by Pauldelbrot ( most common)^{[52]}
 heuristic using interval arithmetic developed by knighty^{[53]}
How to choose reference point ( by Claude):
 simple method: "take the first reference to be the center of the image, and correct any glitches that result (including those resulting from the reference escaping too early) by adding more references within the glitches, recalculating only those pixels that need it. A simple approach can still yield accurate results, albeit in less than optimal time"
 " trying periodic points (the nuclei of the minibrot islands deep in the set) and preperiodic points (the Misiurewicz points at the centers of spirals), both of which can be found by Newton's method (finding their (pre)periods is a bit harder, but not impossible). Higherperiod "structural" minibrot nuclei seem to be the most favoured as they are relatively easy to find while also emitting fewer glitched pixels than lower period nuclei"
graf[edit]
Dessin d'enfant[edit]
Tree[edit]
Farey tree[edit]
Farey tree = Farey sequence as a tree
Hubbard tree[edit]
"Hubbard trees are finite planar trees, equipped with selfmaps, which classify postcritically finite polynomials as holomorphic dynamical systems on the complex plane." ^{[54]}
Invariants[edit]
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[edit]
Dynamical invariants = invariants of the dynamical system
 periodic points
 fixed point
 invariant curve
 periodic ray
 external
 internal
 periodic ray
Dynamical Invariants Derived from Recurrence Plots^{[55]}
Interval[edit]
a partition of an interval into subintervals
 Markov paritition^{[56]}
Iteration[edit]
Itinerary[edit]
is an itinerary of point x under the map f relative to the paritirtion.
It is a rightinfinite sequence of zeros and ones ^{[57]}
where
Examples :
For rotation map and invariant interval ( circle ) :
one can compute :
and split interval into 2 subintervals ( lower circle paritition):
then compute s according to it's relation with critical point :
Itinerary can be converted^{[58]} to point
Magnitude[edit]
 magnitude of the point ( complex number in 2D case) = it's distance from the origin^{[59]}
 radius is the absolute value of complex number ( compare to arguments or angle)
Map[edit]
types[edit]
 The map f is hyperbolic if every critical orbit converges to a periodic orbit.^{[60]}
Complex quadratic map[edit]
Forms[edit]
c form : [edit]
quadratic map^{[61]}
 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 :^{[62]}
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^{[63]}
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) )$
Circle map[edit]
Circle map ^{[64]}
 irrational rotation^{[65]}
Doubling map[edit]
definition ^{[66]}
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 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^{[67]}
 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.".^{[68]} 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.
Feigenbaum map[edit]
"the Feigenbaum map F is a solution of CvitanovicFeigenbaum equation"^{[69]}
First return map[edit]
 definition ^{[70]}
 video : Intro to Poincare map (Poincaré), the first return map. This map helps us determine the stability of a limit cycle using the eigenvalues (Floquet multipliers) associated with the map.
"In contrast to a phase portrait, the return map is a discrete description of the underlying dynamics. .... A return map (plot) is generated by plotting one return value of the time series against the previous one "^{[71]}
"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)
"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." ^{[72]}
"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 ndimensional spaces"^{[73]}
Multiplier map[edit]
Multiplier map associated with hyperbolic component
 gives an explicit uniformization of hyperbolic component by the unit disk :
In other words it maps hyperbolic component H to unit disk D.
It maps point c from parameter plane to point b from reference plane:
where:
 c is a point in the parameter plane
 b is a point in the reference plane. It is also internal coordinate
 is a multiplier map
Multiplier map is a conformal isomorphism.^{[74]}
It can be computed using :
Riemann map[edit]
Riemann mapping theorem^{[75]} says that every simply connected subset U of the complex number plane can be mapped to the open unit disk D
where:
 D is a unit disk
 f is Riemann map ( function)
 U is subset of complex plane
Examples
 multiplier map on the parameter plane
 Böttcher coordinates
 on the parameter plane the Riemann map for the complement of the Mandelbrot set
 on dynamic plane^{[76]}
 for the Fatou component containing a superattracting fixed point for a rational map^{[77]}
 a Riemann map for the complement of the filled Julia set of a quadratic polynomial with connected Julia : "The Riemann map for the central component for the Basilica was drawn in essentially the same way, except that instead of starting with points on a big circle, I started with sample points on a circle of small radius (e.g. 0.00001) around the origin." Jim Belk
function:
 explicit formula ( only in simple cases)
 numerical aproximation ( in most of the cases)^{[78]}
 Zipper
 " Thurston and others have done some beautiful work involving approximating arbitrary Riemann maps using circle packings. See Circle Packing: A Mathematical Tale by Stephenson."
 " To some extent, constructing a Riemann map is simply a matter of constructing a harmonic function on a given domain (as well as the associated harmonic conjugate), subject to certain boundary conditions. The solution to such problems is a huge topic of research in the study of PDE's, although the connection with Riemann maps is rarely mentioned." Jim Belk^{[79]}
PDE's approach to construct a Riemann map explicitly on a given domain D
 First, translate the domain so that it contains the origin.
 Next, use a numerical method to construct a harmonic function F satisfying
for all , and let
Then
 and is harmonic
so:
 R is the radial component (i.e. modulus) of a Riemann map on D.
 The angular component can now be determined by the fact that its level curves are perpendicular to the level curves of R, and have equal angular spacing near the origin."
See commons : Category:Riemann_mapping
Rotation map[edit]
Rotation map describes counterclockwise rotation of point thru turns on the unit circle :
It is used for computing :
Shift map[edit]
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 (onesided binary left shift ) acts on onesided infinite sequence of binary numbers by
It just drops first digit of the sequence.
If we treat sequence as a binary fraction :
then shift map = the dyadic transformation = dyadic map = bit shift map= 2x mod 1 map = Bernoulli map = doubling map = sawtooth map
and "shifting N places left is the same as multiplying by 2 to the power N (written as 2N)"^{[80]} ( operator << )
In Haskell:
shift k = genericTake q . genericDrop k . cycle  shift map
See also:
Multiplier[edit]
Multiplier of periodic zpoint :^{[81]}^{[82]}
 "The value of is the same at any point in the orbit of a: it is called the multiplier of the cycle."^{[83]}
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
Normalize[edit]
 " normalize this vector so it has modulus one " A Cheritat
Number[edit]
complex number[edit]
 numerical value : x+y*i
 vector from origin to point ( x,y)
 point (x,y) od 2D Cartesion plain
Rotation number[edit]
The rotation number^{[84]}^{[85]}^{[86]}^{[87]}^{[88]} 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 ^{[89]}
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
Winding number[edit]
 of the map ( iterated function)^{[90]}^{[91]}
 "the winding number of the dynamic ray at angle a around the critical value, which is defined as follows: denoting the point on the dynamic aray at potential t greater or equal to zero by zt and decreasing t from +infinity to 0, the winding number is the total
change of arg(zt  c) (divided by 2*Pi so as to count in full turns).Provided that the critical value is not on the dynamic ray or at its landing point, the winding number is welldefined and finite and depends continuously on the parameter. " DIERK SCHLEICHER ^{[92]}
 of the curve ^{[93]}
Orbit[edit]
Orbit is a sequence of points^{[94]}
 phase space trajectories of dynamical systems
 The orbit of periodic point is finite and it is called a cycle.
Backward[edit]
Critical[edit]
Forward orbit^{[95]} of a critical point^{[96]}^{[97]} is called a critical orbit. Critical orbits are very important because every attracting periodic orbit^{[98]} attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.^{[99]}^{[100]} ^{[101]}
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 (complexparts z))
(+ repeats 1)
0)))))
Here are images:
 images of critical orbitsat commons
 by Mike Croucher^{[102]}
 Chris King ^{[103]}
 Kerry Mitchel: cardioidboundaryorbits
 list:
 images by Conan written in Rainbow
Forward[edit]
Homoclinic / heteroclinic[edit]
Inverse[edit]
Inverse = Backward
periodic[edit]
skipped[edit]
 set containing first n iterations of initial point without initial point and its k iterations
 number of elements = n  k
It is used in the average colorings
truncated[edit]
 set containing initial point and first n iterations of initial point
 number of elements = n+1
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. " ^{[104]}
 parameter of the function
Paritition[edit]
 Markov
Period[edit]
Period of point under the iterarted function f is the smallest positive integer value p for which this equality
holds is the period^{[105]} of the orbit.^{[106]}
is a point of periodic orbit ( limit cycle ) .
More is here
Perturbation[edit]
 Perturbation technque for fast rendering the deep zoom images of the Mandelbrot set^{[107]}
 perturbation of parabolic point ^{[108]}
 use perturbation theory to approximate the solutions of the differential equations
Plane[edit]
Planes ^{[109]}
Douady’s principle : “sow in dynamical plane and reap in parameter space”.
Dynamic plane[edit]
 zplane for fc(z)= z^2 + c
 zplane for fm(z)= z^2 + m*z
Parameter plane[edit]
See :^{[110]}
Types of the parameter plane :
 cplane ( standard plane )
 exponential plane ( map) ^{[111]}^{[112]}
 flatten' the cardiod ( unroll ) ^{[113]}^{[114]} = "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)^{[115]}
 transformations ^{[116]}
Points[edit]
Bandmerging[edit]
the bandmerging points are Misiurewicz points^{[117]}
Biaccessible[edit]
If there exist two distinct external rays landing at point we say that it is a biaccessible point.^{[118]}
Center[edit]
Nucleus or 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." ^{[119]}
Synonyms :
 Nucleus of a MuAtom ^{[120]}
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." ^{[121]}
Critical[edit]
A critical point^{[122]} 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.^{[123]}
Cut[edit]
Cut point k of set S is a point for which set Sk is dissconected ( consist of 2 or more sets).^{[124]} 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.^{[125]} Cut rays can be used to construct puzzles.
Cut angle is an angle of cut ray.
fixed[edit]
Periodic point when period = 1
Feigenbaum[edit]
The Feigenbaum Point^{[126]} 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)^{[127]}
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^{[128]} when the magnification increases by 4.6692 (the Feigenbaum Constant) and period is doubled each time^{[129]}

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.
" a "Feigenbaum point" (an infinitely renormalizable parameter of bounded type, such as the famous Feigenbaum value which is the limit of the period2 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 RempeGillen^{[130]}
Fibonacci[edit]
Fibonacci point^{[131]} ^{[132]}^{[133]}
infinity[edit]
The point at infinity ^{[134]}" 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 RempeGillen^{[135]}
Misiurewicz[edit]
Misiurewicz point^{[136]} = " parameters where the critical orbit is preperiodic.
Examples are:
 bandmerging points of chaotic bands (the separator of the chaotic bands Bi−1 and Bi )^{[137]}
 the branch points
 tips in the Mandelbrot set ( tips of the midgets ) ^{[138]}
Characteristic Misiurewicz pointof the chaotic band of the Mandelbrot set is :^{[139]}
 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
MyrbergFeigenbaum[edit]
MF = the MyrbergFeigenbaum point is the different name for the Feigenbaum Point.
Parabolic point[edit]
parabolic points : this occurs when two singular points coallesce in a double singular point (parabolic point)^{[140]}
"the characteristic parabolic point (i.e. the parabolic periodic point on the boundary of the critical value Fatou component) of fc"^{[141]}
Periodic[edit]
Point z has period p under f if :
In other words point is periodic
See also:
 fixed point
 stability of periodic point
 attracting
 repelling
 indifferent
 multiplier of periodic cycle
Pinching[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 ^{[142]}
postcritical[edit]
A postcritical point is a point
where is a critical point.^{[143]}
precritical[edit]
precritical points, i.e., the preimages of 0
reference point[edit]
Reference point of the image:
 its orbit ( reference orbit) is computed with arbitrary precision and saved
 orbits of the other points of the image ( noreference points) are computed from reference orbit using standard precision ( with hardware floating point numbers ) = faster then using arbitrary precision
root[edit]
The root point :
 has a rotational number 0
 it is a biaccesible point ( landing point of 2 external rays )
singular[edit]
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 nonessential 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 nonessential singularity itself has a nonessential 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 multivalued function, such as or being defined within a certain limited domain so that the function can be made singlevalued 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 and for ) which are fixed in place.
Portrait[edit]
orbit portrait[edit]
types[edit]
There are two types of orbit portraits: primitive and satellite.^{[144]} 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.
Precision[edit]
Precision of :
 data type used for computation. Measured in bits (width of significant ( fraction) = number of binary digits) or in decimal digits
 input values
 result ( number of significant figures )
See :
 Numerical Precision : " Precision is the number of digits in a number. Scale is the number of digits to the right of the decimal point in a number. For example, the number 123.45 has a precision of 5 and a scale of 2."^{[145]}
 error ^{[146]}
Processes or transformations and phenomenona[edit]
Aliasing and antialising[edit]
Contraction and dilatation[edit]
 the contraction z → z/2
 the dilatation z → 2z.
Discretizations[edit]
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)^{[147]}
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.^{[148]}
Normalization[edit]
Renormalization[edit]
"to any quadratic map f we can associate a canonical sequence of periods p1 < p2 <... for which f is renormalizable.
Depending on whether the sequence is:
 empty
 finite
 infinite
the map f is called respectively:
 nonrenormalizable
 at most finitely renormalizable
 infinitely renormalizable" ^{[149]}
Tuning[edit]
Uniformization[edit]
Vectorisation[edit]
Radius[edit]
Radius of complex number[edit]
The absolute value or modulus or magnitude or radius of a complex number
Conformal radius[edit]
Conformal radius of Siegel Disk ^{[151]}^{[152]}
Escape radius ( ER)[edit]
Escape radius ( ER ) or bailout value is a radius of circle centered at origin ( z=0). This set is used as a target set in the bailout test ( escape time method = ETM )
Minimal Escape Radius should be grater or equal to 2 :
Better estimation is :^{[153]}^{[154]}
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
See also : the N2 rule^{[155]}
Sequences[edit]
A sequence is an ordered list of objects (or events).^{[156]}
A series is the sum of the terms of a sequence of numbers.^{[157]} Some times these names are not used as in above definitions.
Orbit[edit]
Orbit can be:
Set[edit]
Continuum[edit]
definition^{[158]}
Component[edit]
Components of parameter plane[edit]
muatom , ball, bud, bulb, decoration, lake and lakelet.^{[159]}
Islands[edit]
Names :
 mini Mandelbrot set
 'baby'Mandelbrot set
 island mumolecules = embedded copy of the Mandelbrot Set^{[160]}
 Bug
 Island
 Mandelbrotie
 Midget
List of islands :
 http://mrob.com/pub/mudata/largestislands.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/featuredatabase.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
Primitive and satellite[edit]
"Hyperbolic components come in two kinds, primitive and satellite, depending on the local properties of their roots." ^{[161]}
 primitive ( nonsatellite)
 the root of component is not on the boundary of another component = "it was born from another hyperbolic component by the period increasing bifurcation"^{[162]}
 ones that have a cusp likes the main cardioid, when the the little Julia sets are disjoint ^{[163]}
 satellite
 ones that don't have a cusp^{[164]}
 it's root is on the boundary of another hyperbolic component ^{[165]}
 when the little Julia sets touch at their βfixed
point
primare[edit]
Child (Descendant ) and the parent ( ancestor)[edit]
 ancestor of hyperbolic componnet
 descendant of hyperbolic component = child ^{[166]}
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." ^{[167]}
A hyperbolic component is narrow if it contains no component of equal or lesser period in its wake ^{[168]}
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[edit]
p/qlimb is a part of Mandelbrot set contained inside p/qwake
For every rational number , where p and q are relatively prime, a hyperbolic component of period q bifurcates from the main cardioid. The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called the p/qlimb. Computer experiments suggest that the diameter of the limb tends to zero like . The best current estimate known is the Yoccozinequality, which states that the size tends to zero like .
A periodq limb will have q − 1 "antennae" at the top of its limb. We can thus determine the period of a given bulb by counting these antennas.
In an attempt to demonstrate that the thickness of the p/qlimb is zero, David Boll carried out a computer experiment in 1991, where he computed the number of iterations required for the series to converge for z = ( being the location thereof). As the series doesn't converge for the exact value of z = , the number of iterations required increases with a small ε. It turns out that multiplying the value of ε with the number of iterations required yields an approximation of π that becomes better for smaller ε. For example, for ε = 0.0000001 the number of iterations is 31415928 and the product is 3.1415928.^{[169]}
shrub[edit]
"what emerges from MyrrbergFeigenbaum point is what we denominate a shrub due to its shape" M Romera
Wake[edit]
p/qwake 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 :
 Combinatorial algorithm = Devaney's method
 book program by Claude HeilandAllen
 wake function from program Mandel by Wolf Jung
p/qSubwake of W is a wake of a p/qsatellite component of W
wake is named after:
 rotation number p/q ( as above)
 angles of external rays landing in it's root point : "If two Mrays land at the same point we denote by wake the component of which does not contain 0."^{[170]}
Components of dynamical plane[edit]
In case of Siegel disc critical orbit is a boundary of component containing Siegel Disc.
dendrit[edit]
"Complex 1variable polynomials with connected Julia sets and only repelling periodic points are called dendritic."^{[171]}
Domain[edit]
Domain in mathematical analysis it is an open connected set
Jordan domain[edit]
"A Jordan domain^{[172]} 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." ^{[173]}
Dwell bands[edit]
"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). " ^{[174]} Other names:
 level sets of integer escape time
Flower[edit]
Invariant[edit]
"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" ^{[175]}
Examples :
 invariant point = fixed point
 invariant cycle = periodic point
 invariant circle
 petal = invariant planar set
Level set[edit]
 a level set of a realvalued function f^{[176]} ( see also dwell band)
 Level set methods (LSM)
in case of :
 dynamic plane
 integer escape time
 target set : exterior of the circle ( used in the escaping test )
level set is defined :
Boundaries of level sets ( lemniscates) are
Planar set[edit]
a nonseparating planar set is a set whose complement in the plane is connected.^{[177]}
Sepal[edit]
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 : petal[edit]
In the parabolic case target set shoul be iside 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 ).
smooth[edit]
smooth = changing without visible (noticeable) edges
use:
 smooth gradient
similar:
 conitnuous
compare:
 discrete
Surgery[edit]
 surgery in differential topology ^{[178]}
Links:
Test[edit]
Bailout test or escaping test[edit]
It is used to check if point z on dynamical plane is escaping to infinity or not.^{[179]} It allows to find 2 sets :
 escaping points ( it should be also the whole basing of attraction to infinity)^{[180]}
 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).^{[181]}
The output of test can be :
 boolean ( yes/no)
 integer : integer number (value of the last iteration)
Types of bailout test:
Attraction test[edit]
References[edit]
 ↑ Rational Maps with Clustering and the Mating of Polynomials by Thomas Joseph Sharland
 ↑ Topics from OneDimensional Dynamics by Karen M. Brucks,Henk Bruin. page 265 exercise 14.2.12
 ↑ muency  internal angle ( the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872016. )
 ↑ internal angle from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872017
 ↑ 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
 ↑ Matcont  is a Matlab software project for the numerical continuation and bifurcation study of continuous and discrete parameterized dynamical systems. Leaders of the project are Willy Govaerts (Gent,B) and Yuri A. Kuznetsov (Utrecht,NL).
 ↑ geometry by Dr. Carol JVF Burns
 ↑ What is a Curve ?
 ↑ 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
 ↑ Escape lines versus equipotential lines in the Mnadelbrot set by M. Romera, Pastor G , D. de la Guía, Montoya
 ↑ The Computation of Invariant Circles of Maps Article in Physica D Nonlinear Phenomena 16(2):243251 · June 1985 DOI: 10.1016/01672789(85)900612 1st I.G. Kevrekidis
 ↑ A NewtonRaphson method for numerically constructing invariant curves Marty, Wolfgang
 ↑ Numerical Approximation of Rough Invariant Curves of Planar Maps Article in SIAM Journal on Scientific Computing 25(1) · September 2003 DOI: 10.1137/S106482750241373X K. D. Edoh and Jens Lorenz
 ↑ SIAM J. Sci. and Stat. Comput., 8(6), 951–962. (12 pages) A New Algorithm for the Numerical Approximation of an Invariant Curve Published online: 14 July 2006 Keywords invariant manifold, polygonal approximation AMS Subject Headings 65L99, 65H10, 34C40 Publication Data ISSN (print): 01965204 ISSN (online): 21683417 Publisher: Society for Industrial and Applied Mathematics M. van Veldhuizen
 ↑ Escape lines versus equipotential lines in the Mnadelbrot set by M. Romera, Pastor G , D. de la Guía, Montoya
 ↑ 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
 ↑ wikipedi : dense_set
 ↑ mathoverflow question : isthereanalmostdensesetofquadraticpolynomialswhichisnotintheinte/254533#254533
 ↑ fractalforums : denseimage
 ↑ A Cheritat wiki : see image showing gammacorrect downscale of dense part of Mandelbropt set
 ↑ fractal forums : pathfindinginthemandelbrotset/
 ↑ A Cheritat wiki : Mandelbrot_set  Following_the_derivative
 ↑ wikipedia : discretization
 ↑ mathinsight : from_discrete_to_continuous_dynamical_systems
 ↑ Symbolic Dynamics of Quadratic Polynomials by H. Bruin and D. Schleicher
 ↑ Symbolic Dynamics and Rotation Numbers J. J. P. Veerman Phys. 13A, 1986, 543576.
 ↑ Symbolic Dynamics of OrderPreserving Orbits J. J. P. Veerman Phys. 29D, 1987, 191201.
 ↑ Walter Bergweiler : A gallery of complex dynamics pictures.
 ↑ mathoverflow question : whatsanaturalcandidateforananalyticfunctionthatinterpolatesthetower/43003
 ↑ A Cheritat wiki : Mandelbrot_set  Interior_detection_methods
 ↑ MAT335H1F Lecture Notes by Burbulla (Chapter 11, 12 and 13 )
 ↑ SchwarzianDerivativePoster
 ↑ Germ in wikipedia
 ↑ math.stackexchange question : isthereanydifferencebetweenmappingandfunction
 ↑ Iterated function (map) in wikipedia
 ↑ evolution function
 ↑ the discrete nonlinear dynamical system
 ↑ Connectivity of Julia sets of Newton maps: A unified approach by K. Baranski N. Fagella X. Jarque B. Karpinska
 ↑ A Beginners’ Guide to Resurgence and Transseries in Quantum Theories Gerald Dunne
 ↑ dinkydauset at deviantar :PerturbationfortheMandelbrotset450766847
 ↑ math.stackexchange question: selectingreferenceorbitforfractalrenderingwithperturbationtheory
 ↑ fractalforums " pertubationtheoryglitchesimprovement
 ↑ fractalforums : *continued*superfractalthingarbitraryprecisionmandelbrotsetrenderinginja/msg91505/#msg91505
 ↑ Dessins d’enfants and Hubbard trees by Kevin M. Pilgrim
 ↑ N. Marwan, M. C. Romano, M. Thiel, J. Kurths: Recurrence Plots for the Analysis of Complex Systems, Physics Reports, 438(56), 237329, 2007.
 ↑ math.stackexchange question : definitionofmarkovpartition
 ↑ Structure of Inverse Limit Spaces of Tent Maps with Nonrecurrent Critical Points by Brian Raines and Sonja Stimac
 ↑ Bifurcation structures in maps of Henon type by Kai T Hansen and Predrag Cvitanovic
 ↑ wikipedia: Magnitude in mathematics
 ↑ Hyperbolic Components by John Milnor
 ↑ Complex quadratic map in wikipedia
 ↑ Michael Yampolsky, Saeed Zakeri : Mating Siegel quadratic polynomials.
 ↑ Mandel: software for real and complex dynamics by Wolf Jung
 ↑ threecoolfactsaboutrotationsofthecircle by David Richeson
 ↑ irrationalrotationsofthecircleandbenfordslaw by David Richeson
 ↑ wikipedia : Dyadic transformation
 ↑ lavaurs' algorithm in Haskell with SVG output by Claude HeilandAllen
 ↑ SYMBOLIC DYNAMICS AND SELFSIMILAR GROUPS by VOLODYMYR NEKRASHEVYCH
 ↑ The measure of the Feigenbaum Julia set by Artem Dudko and Scott Sutherland
 ↑ Poincaré map
 ↑ General principles of chaotic dynamics by P.B. Persson , C.D. Wagner
 ↑ Continuous time and discrete time dynamical systems by Shaun Bullett
 ↑ Continuous time and discrete time dynamical systems by Shaun Bullett
 ↑ Conformal Geometry and Dynamics of Quadratic Polynomials Mikhail Lyubich
 ↑ wikipedia: Riemann mapping theorem
 ↑ A THOMPSON GROUP FOR THE BASILICA by JAMES BELK AND BRADLEY FORREST
 ↑ math stackexchange question: explicitriemannmappings
 ↑ mathoverflow question: complexfunctionformappingacircletoasuperellipse
 ↑ math.stackexchange question: explicitriemannmappings
 ↑ binary_shift_left
 ↑ Multiplier at wikipedia
 ↑ Internal angles and multipliers from Fractal Geometry Yale University Michael Frame, Benoit Mandelbrot (19242010), and Nial Neger September 3, 2017
 ↑ A Cheritat wikidraw : Mandelbrot_set#Following_the_derivative
 ↑ wikipedia : Rotation number
 ↑ rotation number From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872016
 ↑ scholarpedia : Rotation_theory
 ↑ The Fractal Geometry of the Mandelbrot Set II. How to Count and How to Add Robert L. Devaney
 ↑ An Introduction to Rotation Theory Prize winner, DSWeb Student Competition, 2007 By Christian Kue
 ↑ Complex systems simulation Curso 20122013 by Antonio Giraldo and María Asunción Sastre
 ↑ Weisstein, Eric W. "Map Winding Number." From MathWorldA Wolfram Web Resource. http://mathworld.wolfram.com/MapWindingNumber.html
 ↑ wikipedia : Rotation_number
 ↑ RATIONAL PARAMETER RAYS OF THE MANDELBROT SET by Dierk Schleicher
 ↑ https://plus.maths.org/content/windingnumberstopographyandtopologyii
 ↑ wikipedia : Orbit_(dynamics)
 ↑ 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
 ↑ Khan Academy : Mandelbrot Spirals 2
 ↑ Complex Power Towers (Or ‘mucking around with Mathematica’) by Mike Croucher
 ↑ /DarkHeart by Chris King
 ↑ Ouadraticlike maps and Renormalization by Nuria Fagella
 ↑ Peiod From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872015.
 ↑ scholarpedia : Periodic Orbit for a Map
 ↑ Perturbation for the Mandelbrot set by DinkydauSet, Apr 28, 2014, 3:46:13 PM
 ↑ PARABOLIC IMPLOSION A MINICOURSE ARNAUD CHERITAT
 ↑ 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
 ↑ Nucleus  From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872015.
 ↑ 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
 ↑ mathoverflow question : isthereawaytofindregionsofdepthinthemandelbrotsetotherthansimply?rq=1
 ↑ Fractalforums : fibonacciandthemandelbrotset
 ↑ Parameter scaling for the Fibonacci point by Leroy Wenstrom
 ↑ The Fibonacci unimodal map by Mikhail Lyubich, John W. Milnor
 ↑ [w:Point at infinityPoint at infinity in wikipedia]
 ↑ Mathoverflow question : Attractive Basins and Loops in Julia Sets
 ↑ wikipedia : Misiurewicz point
 ↑ Symbolic sequences of onedimensional quadratic map points by G Pastor, Miguel Romera, Fausto Montoya Vitini
 ↑ mathoverflow question : Is there a way to find regions of depth in the Mandelbrot set other than simply poking around?
 ↑ 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
 ↑ The bifurcation diagram of cubic polynomial vector fields on CP1 by Christiane Rousseau
 ↑ A rigidity result for some parabolic germs by Luna Lomonaco, Sabyasachi Mukherjee
 ↑ 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
 ↑ stackoverflow question : howdoiinterpretprecisionandscaleofanumberinadatabase
 ↑ takingtheerroroutoftheerrorfunction by Fredrik Johansson
 ↑ Airplane primitive parabolic implosion by Wolf Jung
 ↑ CANTOR BOUQUETS, EXPLOSIONS, AND KNASTER CONTINUA: DYNAMICS OF COMPLEX EXPONENTIALS by Robert L. Devaney Publicacions Matematiques, Vol 43 (1999), 27–54.
 ↑ Baby Mandelbrot sets, Renormalization and MLC Mikhail Lyubich
 ↑ Tuning From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872014.
 ↑ 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
 ↑ The Mandelbrot Set and Julia Sets Combinatorics in the Mandelbrot Set  The 1/n2 rule, and deviations from it
 ↑ wikipedia : Sequence
 ↑ wikipedia : series
 ↑ wikipedia : Continuum in set theory
 ↑ Muatom From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872013.
 ↑ Island MuMolecule by Robert P. Munafo, 2012 Aug 18.
 ↑ Internal addresses in the Mandelbrot set and Galois groups of polynomials by Dierk Schleicher, page 31
 ↑ notes by Lyubich
 ↑ Satellite copies of the Mandelbrot set by Luna Lomonaco
 ↑ mathoverflow : preciselocationofthemandelbrotbulbattachmenttothemaincardioid
 ↑ ON BIACCESSIBLE POINTS OF THE MANDELBROT SET by SAEED ZAKERI
 ↑ Child From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872013.
 ↑ Surgery in Complex Dynamics by Carsten Lunde Petersen, online paper
 ↑ Internal addresses in the Mandelbrot set and irreducibility of polynomials by Dierk Schleicher
 ↑ Gary William Flake, The Computational Beauty of Nature, 1998. p. 125. ISBN 9780262561273.
 ↑ Local properties of the Mandelbrot set at parabolic points by Tan Lei
 ↑ LAMINATIONAL MODELS FOR SOME SPACES OF POLYNOMIALS OF ARBITRARY DEGREE by ALEXANDER BLOKH, LEX OVERSTEEGEN, ROSS PTACEK, AND VLADLEN TIMORIN
 ↑ wikipedia : Carathéodory's theorem (conformal mapping)
 ↑ The intrinsic geometry of a Jordan domain by Richard L. Bishop
 ↑ fractalforums : binarydecompositionandexternalangles by Claude
 ↑ Norman Lebovitz : Textbook for Mathematics 27300
 ↑ wikipedia : Level set
 ↑ 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
 ↑ wikipedioa: Surgery theory
 ↑ Fractus doc by Richard Rosenman
 ↑ wikipedia : Escaping set
 ↑ fractint doc : bailout