Fractals/Iterations in the complex plane/Parameter plane
Structure of parameter planeEdit
The phase space of a quadratic map is called its parameter plane. Here:
 is constant
 is variable
There is no dynamics here. It is only a set of parameter values. There are no orbits on the parameter plane.
The parameter plane consists of :
 The Mandelbrot set
 The bifurcation locus = boundary of Mandelbrot set
 Bounded hyperbolic components of the Mandelbrot set = interior of Mandelbrot set ^{[1]}
Structure of parameter plane from programmers point of view:
 exterior of Mset
 Each component is surrounded by an atom domain ( disc with radius 4 times bigger, for cardioids radius has about the square root of the size).
 Each component has a nucleus at its center, which has a periodic orbit containing 0.
curvesEdit
The Mandelbrot set contains smooth curves:
 the intersection with the real axis M ∩ R = [−2, 1/4] = real slice of Mandelbrot set
 the main cardioid of M, which is the set of parameters c for which fc has an attracting or indifferent fixed point (of course, this is smooth except at the cusp c = 1/4).^{[2]}
 boundary of period 2 hyperbolic componenet, which is a circle
real sliceEdit
" each perioddoubling cascade superstable orbit ... generate the corresponding chaotic band and the Misiurewicz point that separates the chaotic bands and "^{[3]}
where:
 B is a chaotic region from 2 to MF
 = Feigenbaum point = the MyrbergFeigenbaum
 A is a periodic region : from MF to 1/4
 is the precedes symbol. The binary relationship: "x precedes y" is written: . It is used to distinguish other orders from total orders.
chaotic regionEdit
where:
 is chaotic band
See also
 separators between bands
 It is a part of Sharkovsky ordering
periodic regionEdit
 2^n = the powers of 2 in decreasing order, ending with 2^0 = 1."^{[4]}
named parts of Mandelbrot setEdit
Valley:
 double spiral
 spindle
 seahorse valley/coast ( or Seahorse Valley East ) ^{[5]}
 main cardioid seahorse valley = Gap between the head ( period 2 component) and the body (or shoulder = main cardioid). Particularly the upper one part.^{[6]}
 disk 3 seahorse valley = Gap between period 1 and period 3
 Elephant Valley = Gap near cusp of main cardioid. Here the antennae resemble the trunks of elephants^{[7]}
 elephant coast = boundary of main cardioid near cusp
 Scepter Valley = gap between period 2 and period 4 component, Also known as Seahorse Valley West or Sceptre Valley. "Scepter Valley" where double spirals have scepters coming (echoing the spindle) coming off of all their tips.^{[8]} ^{[9]}
 double spiral valley =
 double spiral coast = boundary of period 2 component near main cardioid
 triple spiral coast ( valley)  boundary of period 3 componnet near period 1 cardioid
Parts
 Peacock eye = shrub ( decoration) from Seahorse valley = Principal Misiurewicz point with all branches
 seahorse
 elephant

Gap between the head and the body = seahorse valley

Doublespirals on the left, seahorses on the right

Fragment of Mandelbrot set called the elephant valley
Mini Mandelbrot setsEdit
Names:
 midget
 mini mandelbrot set
 baby Mandelbrots
 island = island mumolecule = island muunit ^{[10]}
 primitive component
Parameter rays of mini Mandelbrot sets ^{[11]}
" a saddlenode (parabolic) periodic point in a complex dynamical system often admits homoclinic points, and in the case that these homoclinic points are nondegenerate, this is accompanied by the existence of infinitely many baby Mandelbrot sets converging to the saddlenode parameter value in the corresponding parameter plane." Devaney ^{[12]}
See also
 How to find the angles of external rays that land on the root point of any Mandelbrot set's component which is not accesible from main cardioid ( M0) by a finite number of boundary crossing ?
 distorted islands
Julia island = Embedded Julia Set = virtual Julia SetsEdit
 "A structure comprised of filaments, resembling a Julia set in appearance, which has a higher delta Hausdorff dimension than filaments in the immediately surrounding region. They are sometimes also called Julia islands or virtual Julia Sets." Robert Munafo^{[13]}
 "This location is called Julia island since it looks like a Julia set but it's actually inside Mandelbrot." ^{[14]}
hyperbolic componenetsEdit
order of hyperbolic componenetsEdit
 Myrberg 1963
 Sharkovsky 1964
 Metropolis 1973
 internal adress by Lau and Schleicher 1994
 rotation number by Devaney 1997
 R2 naming system by R Munafo 1999^{[15]}
 orbit portrait by Milnor 2000
 shrubs by M Romero et al 2004 ^{[16]}
shapeEdit
 ON THE SHAPES OF ELEMENTARY DOMAINS OR WHY MANDELBROT SET IS MADE FROM ALMOST IDEAL CIRCLES by V. Dolotin A.morozov
 Algebraic Geometry of Discrete Dynamics. The case of one variable by V.Dolotin, A.Morozov
Parts of parameter planeEdit
 with respect to the Mandelbrot set
 with respect to the wakes
 inside a p/q wake
 outside any wake (???)
Parts of Mandelbrot set according to M Romera et al.:^{[17]}
 main cardioid
 q/p family (= q/p limb)
 periodic part: period doubling cascade of hyperbolic components which ends at the MyrbergFeigenbaum point
 the MyrbergFeigenbaum point
 chaotic part: shrub
Not that her q/p not p/q notation is used
Components:
 wake / limb
 migdet / island / mumolecule^{[18]}
 distorted^{[19]} = warped^{[20]}
 hyperbolic component of Mandelbrot set
Names source
 muency ^{[21]}
 Mandlebrot
How to choose a point from parameter plane ?Edit
 take point ( and check it's properities)
 taking parameter choosing by other people (visual choose)
 clicking on parameter points and see what you have ( random choose)^{[22]}
 computing a point from it's known properties
 For (parabolic point ) choose hyperbolic component ( period, number) and internal angle (= rotation number) then compute c parameter.
 Misiurewicz points ^{[23]}
 see also known regins in ^{[24]}
 morphing
 interesting areas^{[25]}
 zoom
Is parameter tweaking an acquired art or just random chance?
"From my own experience with monocritical polynomials and Lyapunov diagrams, all my images I found purely by chance. For z^2+c e.g. as long as you're in the same hyperbolic component, the shape changes only in the sense, that a fat spiral might become thinner, but the number of arms stays constant. If you move the c value out of that component into another  and if this 2nd component is not directly attached to the first, then, I'm not aware of a direct way of telling what it would look like there. Usually I perform a parameter walk just computing black and white escape images with periodicity. Then for the interesting shapes/periods I apply a color walk with some gradients that looked fine in previous images. But that's more or less guessing. If you want to turn more into the constructinganimage from a vision you have, you might try two articles: genetic algorithm and Leja points" marcm200^{[26]}
Mandelbrot set : z^6+ A*z+c How do you find such intereseting examples ?
" I'm running from time to time an A,cparameter space walk (brute force) in a rather wide grid (2..+2 in ~0,01 or larger steps) for the family z^n+A*z+c, adding a small random dyadic fraction to the 4d coordinates to get variation. Following numerically the orbits of the critical points with a rather high max it of 25000 it's possible to get the number of attracting cycles and their length to some accuaracy level in a decent time. If those A,cparameter pairs pass some filters (mostly sum of length of cycles and diversity) I scan through small overview pictures manually. Then I use interesting A,c pairs (shapewise or from the filter values) and some small deviations from it to compute level 1012 TSA images, as sets wiith similar shapes can show a different dynamical behaviour w.r.t. the level at which interior cells emerge. I'll take the fastest one and see how many cycles can be detected up to level 1819." marcm200^{[27]}
How to move on parameter plane ?Edit

Values of C for each frame evaluates by equation: C=r*cos(a)+i*r*sin(a), where: a=(0..2*Pi), r=0.7885. Thus, parameter С outlines circle with a radius r=0.7885 and a center at origin of the complex plane.


zoom and translation towards Feigenbaum point

c=0

c=1/4

c= 1/4 + 0.05

c = 1/4 + 0.029

c = 1/4 + 0.035
How to choose step of a move ?Edit
 fixed size step / adaptive size step
 minimal size of the move
"For c values that could not be represented accurately by C++ double data type I calculated the images using interval arithmetics with tiny intervals (border values being fractions of 2^25, interval width 2^25) encompassing the published value (as needed for real or imaginary part or both) (values were computed using wolframalpha to multiply large integers). Given is the left (smaller) border value, the right is obtained by adding one." marcm200^{[28]}
What is the reason for : 2^25 ?
"When I started with this article back in March this year, my initial formulas were z^2+c and z^2+c*z. In expanded form with real coordinates: z=(x+i*y) and c=(d+e*i): ( x²y² + d, 2xy + e ) or ( d*x + x²  e*y  y² , e*x + d*y + 2x*y ) To accurately represent a sum, the widest two terms can be apart is 53 bits (mantissa precision for C++ double) and all other must lie in this range. The smallest nonzero value of x,y is "axis range / pixel count", i.e. 4 (escape radius of 2, hence axis 2..+2) divided by 2^refinement level. So for x^2 this goes to 2^26 as the lowest possible value for x. And since d,e are multiplied with x in the 2nd formula, the same goes for d and e. As I do not like to work "on the edge" I used a buffer of 12 bits and came to the lowest value of 2^25 for d,e (and refinement level limit of 27 which is currently outside a reasonable range). For the 1st formula z^2+c, the seed value could go as little as 2^48 (stated in the article) as it is only added. For long double and float128 one could go lower in both formulas, but I haven't explored that."marcm200^{[29]}
Compare
 Numerical calculations and rigorous mathematics
 Reliable Computing, Interval Computations
TypesEdit
 type of the move
 continuous
 discrete = using sequence of points
 type of the curve
 along radial curves :
 escape route
 external ray
 parabolic point = root point = landing point of above external ray
 internal ray which also ends/lands at the parabolic point
 along circular curves :
 equipotentials
 boundaries of hyperbolic components
 internal circular curves
 loops ^{[30]}
 along radial curves :
The dynamics of the polynomials of moving along curves is : ^{[31]}
 for external ray: ”stretch” the dynamic on the basin of infinity. The argument of φPa,b (2a) stay unchanged ( fixed)
 for equipotential: twist the dynamics in the annulus between the Green level curves of the escaping critical point and of the critical value. The modulus of φPa,b (2a is fixed
ExamplesEdit

point c moves along boundary of main cardioid toward c=0.75 ( root point of period 2 component of Mandelbrot set) using a sequence
External ray
 From Cantor to Semihyperbolic Parameters along External Rays^{[32]}
 along parameter external ray for angle 9/31 by David Madore : Its external argument is constantly 9/31 and it approaches the bud's root (~ −0.481763 + +0.531657i) exponentially slowly.
 external angle 1/3 by David Madore
Boundary of the component
Others
 morphing
 poincare_halfplane_metric_for_zoom_animation by Claude HeilandAllen
 youtube: Julia sets as C pans over the Mandelbrot set by captzimmo
 youtube : Julia sets about the main cardioid x 1.1 with Mandelbrot set by Thomas Fallon
 youtube: Julia Sets Relative to the Mandelbrot Set by Gary Welz
 you tube : Julia Sets of the Quadratic by Gary Welz
 youtube : Julia set morph around the cardioid / central bulb by blimeyspod
 youtube : Julia set morph / fractal animation  Beyond the Cardioid Perimiter by blimeyspod
 youtube: Julia set morph / fractal animation  Beyond a 2nd Order Bulb by blimeyspod
 Fractals: A tour of Julia Sets by corsec
 shadertoy : Julia  Distance by iq
 Evolving Julia Marco_Gilardi
// glsl code by iq from https://www.shadertoy.com/view/Mss3R8 float ltime = 0.50.5*cos(time*0.12); vec2 c = vec2( 0.745, 0.186 )  0.045*zoom*(1.0ltime);
// glsl code by xylifyx from https://www.shadertoy.com/view/XssXDr vec2 c = vec2( 0.37+cos(iTime*1.23462673423)*0.04, sin(iTime*1.43472384234)*0.10+0.50);
// by Marco Gilardi // https://www.shadertoy.com/view/MllGzB vec2 c = vec2(0.754, 0.05*(abs(cos(0.1*iTime))+0.8));
Escape routeEdit

escape route 1/2
Escape route 0Edit
Escape route for internal angle 0/1
Steps
 nucelus of period 1 component ( c = 0) Fixed point alfa is supperattracting fixed point. Julia set is connected.
 along internal ray 0. Imaginary part of parameter c is zero. 0 < cx < 0.25. Fixed point alfa is attracting fixed point. Julia set is connected.
 parabolic point c = 1/4. Fixed point alfa is parabolic fixed point. Julia set is connected.
 along external ray 0. Imaginary part of parameter c is zero. 0.25 < cx. Fixed point alfa is repelling fixed point. Julia set is disconnected
Here parabolic implosion/ explosion ( from connected to disconnected ) occurs.
In parabolic point child periodic points coincides with parent period points
parameter c  location of c  Julia set  interior  type of critical orbit dynamics  critical point  fixed points  stability of alfa 

c = 0  center, interior  connected  exist  superattracting  atracted to alfa fixed point  fixed critical point equal to alfa fixed point, alfa is superattracting, beta is repelling  r = 0 
0<c<1/4  internal ray 0, interior  connected  exist  attracting  atracted to alfa fixed point  alfa is attracting, beta is repelling  0 < r < 1.0 
c = 1/4  cusp, boundary  connected  exist  parabolic  atracted to alfa fixed point  alfa fixed point equal to beta fixed point, both are parabolic  r = 1 
c>1/4  external ray 0, exterior  disconnected  disappears  repelling  repelling to infinity  both finite fixed points are repelling  r > 1 
Stability r is absolute value of multiplier at fixed point alfa:
c = 0.0000000000000000+0.0000000000000000*I m(c) = 0.0000000000000000+0.0000000000000000*I r(m) = 0.0000000000000000 t(m) = 0.0000000000000000 period = 1 c = 0.0250000000000000+0.0000000000000000*I m(c) = 0.0513167019494862+0.0000000000000000*I r(m) = 0.0513167019494862 t(m) = 0.0000000000000000 period = 1 c = 0.0500000000000000+0.0000000000000000*I m(c) = 0.1055728090000841+0.0000000000000000*I r(m) = 0.1055728090000841 t(m) = 0.0000000000000000 period = 1 c = 0.0750000000000000+0.0000000000000000*I m(c) = 0.1633399734659244+0.0000000000000000*I r(m) = 0.1633399734659244 t(m) = 0.0000000000000000 period = 1 c = 0.1000000000000000+0.0000000000000000*I m(c) = 0.2254033307585166+0.0000000000000000*I r(m) = 0.2254033307585166 t(m) = 0.0000000000000000 period = 1 c = 0.1250000000000000+0.0000000000000000*I m(c) = 0.2928932188134524+0.0000000000000000*I r(m) = 0.2928932188134524 t(m) = 0.0000000000000000 period = 1 c = 0.1500000000000000+0.0000000000000000*I m(c) = 0.3675444679663241+0.0000000000000000*I r(m) = 0.3675444679663241 t(m) = 0.0000000000000000 period = 1 c = 0.1750000000000000+0.0000000000000000*I m(c) = 0.4522774424948338+0.0000000000000000*I r(m) = 0.4522774424948338 t(m) = 0.0000000000000000 period = 1 c = 0.2000000000000000+0.0000000000000000*I m(c) = 0.5527864045000419+0.0000000000000000*I r(m) = 0.5527864045000419 t(m) = 0.0000000000000000 period = 1 c = 0.2250000000000000+0.0000000000000000*I m(c) = 0.6837722339831620+0.0000000000000000*I r(m) = 0.6837722339831620 t(m) = 0.0000000000000000 period = 1 c = 0.2500000000000000+0.0000000000000000*I m(c) = 0.9999999894632878+0.0000000000000000*I r(m) = 0.9999999894632878 t(m) = 0.0000000000000000 period = 1 c = 0.2750000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.3162277660168377*I r(m) = 1.0488088481701514 t(m) = 0.0487455572605341 period = 1 c = 0.3000000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.4472135954999579*I r(m) = 1.0954451150103321 t(m) = 0.0669301182003075 period = 1 c = 0.3250000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.5477225575051662*I r(m) = 1.1401754250991381 t(m) = 0.0797514300099943 period = 1 c = 0.3500000000000000+0.0000000000000000*I m(c) = 1.0000000000000000+0.6324555320336760*I r(m) = 1.1832159566199232 t(m) = 0.0897542589928440 period = 1

center = superattracting

attracting

parabolic

repelling

target set

Internal level sets

binary decomposition

Full tile = binary decomposition and internal level sets
parameter c value  description of c locations  fixed points  Julia set  basins  target set ( petal) 

1/4 < c  point c is on the external ray 0  both fixed points are repelling  disconnected  only ona basin of attraction ( infinity)  
c = 1/4  cusp of main carioid  both fixed points are parabolic ( belong to Julia set)  connected = Cauliflower  circle  
0 < c < 1/4  inside main cardioid, along internal ray 0  Treść komórki  Treść komórki  
c = 0  center of main cardioid  Treść komórki  Treść komórki  
0 < c < 3/4  inside main cardioid, along internal ray 1/2  Treść komórki  Treść komórki  
c = 3/4  root point ( parabolic)  Treść komórki  Treść komórki  
3/4 < c < 1.0  inside period 2 component, along internal ray 0  Treść komórki  Treść komórki  
c = 1.0  center of period 2 component  Treść komórki  Treść komórki  
1/0 < c < 5/4  inside period 2 component, along internal ray 1/2  Treść komórki  Treść komórki  
c = 5/4  root point ( parabolic)  Treść komórki  Treść komórki 
stability index of period 1 points  period 1 points on dynamic plane  period 1 points on parameter plane 

changes from attractive through indifferent to repelling  moves from interior of Kc to its boundary  moves from interior of componetnt of Mset to its boundary 
Escape route 1/3Edit

parameter plane with escape route 1/3

animation from center to parabolic

superattracting = center for period 1

parabolic = boundary

superattracting = center for period 3
Plane typesEdit
Criteria for plane classifications
 fractal formula ( function): cplane, lambdaplane
 plane transformations
There are many different types of the parameter plane^{[33]}^{[34]}
 by function
 plain cplane:
 plain lambda plane where
 by transformations ^{[35]}
 inverted cplane = 1/c plane:
 exponential plane ( map) ^{[36]}^{[37]}^{[38]}
 unrolled plain (flatten' the cardiod = unroll ) ^{[39]}^{[40]} = "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)^{[41]}

cplane

inverted c plane = 1/c plane

plane

Unrolled main cardioid of Mandelbrot set for periods 713

lambda plane

1/lambda plane
Parameter space types by dimensionsEdit
 1D ( 1 real parameter):
 2D ( 1 complex parameter): standard Mandelbrot set, here space is a 2D plane
 4D ( 2 complex parameters) : the family f(z) = z^n+A*z+c by marcm200
 6D ( 3 complex parameters) : six dimensional space of the complex parameters m, b, and d used in the formula f(x)=mx(1x)(x+b)/(x+d) by Valannorton
One can only show 2D slice in the multi dimensional space.
How to describe c point ?Edit
Numerical description
 c value
 Cartesion description
 real part
 imaginary part
 polar description:
 (external or internal ) angle
 ( external or internal) radius, see stability index
 Cartesion description
Symbolic description
 set realtion: Julia set interior / boundary / exterior
Point TypesEdit
point =pixel of parameter plane = c parameter
CriteriaEdit
Criteria for classification of parameter plane points :
 arithmetic properties of internal angle (rotational number) or external angle
 in case of exterior point:
 type of angle : rational, irrational, ....
 preperiod and period of angle under doubling map
 in case of boundary point :
 preperiod and period of external angle under doubling map
 preperiod and period of internal angle under doubling map
 in case of exterior point:
 set properties ( relation with the Mandelbrot set and wakes)
 interior
 boundary
 exterior
 inside wake, subwake
 outside all the wakes, belonging to a parameter ray landing at a Siegel or Cremer parameter,
 geometric properties
 number of external rays that land on the boundary point : tips ( 1 ray), biaccesible, triaccesible, ....
 position of critical point with relation to the Julia set
 Renormalization
ClassificationEdit
There is no complete classification. The "unclassifed" parameters are uncountably infinite, as are the associated angles.
Simple classificationEdit
 exterior of Mandelbrot set
 Mandelbrot set
 boundary of Mandelbrot set
 interior of Mandelbrot set ( hyperbolic parameter)
 centers,
 other internal points ( points of internal rays )
Definitions
 a parameter c in the boundary of Mandelbrot set ∂M is semihyperbolic if the critical point is nonrecurrent and belongs to the Julia set^{[42]}
 A typical example of semihyperbolic parameter is a Misiurewicz point: We say a parameter ˆc is Misiurewicz if the critical point of fcˆ is a preperiodic point.
partial classification of boundary pointsEdit
Classification :^{[43]}
 Boundaries of primitive and satellite hyperbolic components:
 Boundary of M without boundaries of hyperbolic components:
 nonrenormalizable (Misiurewicz with rational external angle and other).
 renormalizable
 finitely renormalizable (Misiurewicz and other).
 infinitely renormalizble (Feigenbaum and other). Angle in turns of external rays landing on the Feigenbaum point are irrational numbers
 non hyperbolic components ( we believe they do not exist but we cannot prove it ) Boundaries of nonhyperbolic components would be infinitely renormalizable as well.
Here "other" has not a complete description. The polynomial may have a locally connected Julia set or not, the critical point may be rcurrent or not, the number of branches at branch points may be bounded or not ...
AlgorithmsEdit
 points ( coordinate)
 compute c from multiplier, period and center (inverse multiplier map)
 compute multiplier from c ( multiplier map)
 dynamics
 Escape time
 DEM/M
 Discrete Velocity of nonattracting Basins and Petals by Chris King
 atom domains
 average distance between random points^{[44]}
 zoom
 combinatorial : tuning
Examples:
 An effective algorithm to compute Mandelbrot sets in parameter planes by A. Garijo, X. Jarque, J. Villadelprat Published 2017
 codegolf SE question: mandelbrotimageineverylanguage
 codegolf SE question: generateamandelbrotfractal ( ASCI)
 rosetta code : Mandelbrot_set
 stackoverflow question: codegolfthemandelbrotset
 stackoverflow questions tagged mandelbrot
 benchmark
ModelsEdit

Topological model of Mandelbrot set( reflects the structure of the object ). Topological model of Mandelbrot set without mini Mandelbrot sets and Misiurewicz points (Cactus model)

Shrub model of Mandelbrot set

Topological model of Mandelbrot set using Lavaurs algorithm up to period 12
Structure of the Mandelbrot set
Size or areaEdit
See alsoEdit
RerferencesEdit
 ↑ Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
 ↑ Generalizations of Douady's magic formula by Adam Epstein, Giulio Tiozzo
 ↑ [Pastor97a] : Harmonic structure of onedimensional quadratic maps by Gerardo Pastor, Miguel Romera, Fausto Montoya Vitini
 ↑ The OnLine Encyclopedia of Integer Sequences : A005408 = The odd numbers: a(n) = 2n+1
 ↑ mandelmap  A detailed map of the Mandelbrot Set, in a beautiful vintage style by Bill Tavis
 ↑ seahorsevalley From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872022
 ↑ mandelbrotlocations by woofractal
 ↑ Mandelbrot Buds and Branches by Timothy Chase
 ↑ Map of the Mandelbrot Set. Copyright © 20052011 Janet Parke
 ↑ the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872018
 ↑ Parameter rays of mini mandelbrot sets
 ↑ Devaney In Global Analysis of Dynamical Systems, ed.: H. Broer, B. Krauskopf, G. Vegter. IOP Publishing (2001), 329338 or Homoclinic Points in Complex Dynamical Systems
 ↑ embedded julia set from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872017.
 ↑ https://www.flickr.com/photos/nonnameavailable/28654921940/
 ↑ https://www.mrob.com/pub/muency/r2namingsystem.html
 ↑ M. Romera et al, Int. J. Bifurcation Chaos 13, 2279 (2003). https://doi.org/10.1142/S0218127403007941 Shrubs in the Mandelbrot Set Ordering
 ↑ SHRUBS IN THE MANDELBROT SET ORDERING by M Romero, G Pastor, G Alvarez, F Montoya
 ↑ mumolecule From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872020.
 ↑ fractalforums.com : howdistortedcanaminibrotbe
 ↑ distribution From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872020
 ↑ MuEncy  The Encyclopedia of the Mandelbrot Set by R Munafo
 ↑ fractalforums : parameteradjustmentartorluck ?
 ↑ interesting c points by Owen Maresh
 ↑ Visual Guide To Patterns In The Mandelbrot Set by Miqel
 ↑ fractalforums : deepzoomingtointerestingareas
 ↑ fractalforums.org : parameteradjustmentartorluck
 ↑ fractalforums.org: juliasetstrueshapeandescapetime
 ↑ fractalforums.org : juliasetstrueshapeandescapetime
 ↑ fractalforums.org : juliasetstrueshapeandescapetime
 ↑ math.stackexchange question: parameterplanedynamicsoffixedpointsandtheirpreimagesforstandardquadra
 ↑ Reading escaping trees from Hubbard trees in Sn by Matthieu Arfeux
 ↑ From Cantor to Semihyperbolic Parameter along External Rays March 2018Transactions of the American Mathematical Society 372(11) DOI: 10.1090/tran/7839 YiChiuan ChenTomoki KawahiraTomoki Kawahira
 ↑ Alternate Parameter Planes by David E. Joyce
 ↑ exponentialmap by Robert Munafo
 ↑ Twisted Mandelbrot Sets by Eric C. Hill
 ↑ muency : exponential map by R Munafo
 ↑ Exponential mapping and OpenMP by Claude HeilandAllen
 ↑ exponential_mapping_with_kalles_fraktaler by Claude HeilandAllen
 ↑ Linas Vepstas : Self Similar?
 ↑ the flattened cardioid of a Mandelbrot by Tom Rathborne
 ↑ Stretching cusps by Claude HeilandAllen
 ↑ From Cantor to Semihyperbolic Parameter along External Rays March 2018Transactions of the American Mathematical Society 372(11) DOI: 10.1090/tran/7839
 ↑ stackexchange : classificationofpointsinthemandelbrotset
 ↑ fractalforums : trickymandelbrotproblem