# Fractals/Iterations in the complex plane/Parameter plane

Parameter plane
* structure
* algorithms
* models

# Structure of parameter plane

The phase space of a quadratic map is called its parameter plane. Here:
* $z0=z_{cr}\,$ is constant
* $c\,$ 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
Structure of parameter plane from programmers point of view:
* exterior of M-set
* 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.
Parts of parameter plane
* components ( incuding islands)
* curves
* points

## curves or paths

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).
* boundary of period 2 hyperbolic componenet, which is a circle
Paths:
* external rays
* internal rays
* escape route

## real slice

" each period-doubling cascade superstable orbit ... generate the corresponding chaotic band $B_{n}$ and the Misiurewicz point $m_{n}$ that separates the chaotic bands $B_{n=1}$ and $B_{n}$ "
$B\prec MF\prec A$
where:
* B is a chaotic region from -2 to MF
* $c=MF=m_{\infty }=b_{\infty }=-1.4011551890......$ = Feigenbaum point = the Myrberg-Feigenbaum
* A is a periodic region : from MF to 1/4
* $\prec$ is the precedes symbol. The binary relationship: "x precedes y" is written: $x\prec y$. It is used to distinguish other orders from total orders.

### chaotic region

$B_{0}\prec B_{1}\prec B_{2}\prec ...\prec B_{\infty }$
where:
* $B_{m}$ is chaotic band $B_{m}=(2n+1)\cdot 2^{m}$
See also
* separators between bands
* It is a part of Sharkovsky ordering

### periodic region

period doubling cascade
* 2^n = the powers of 2 in decreasing order, ending with 2^0 = 1."
$2^{\infty }\prec ...\prec 2^{2}\prec 2^{1}\prec 2^{0}$

## named parts of Mandelbrot set

Valley:
* double spiral
* spindle
* seahorse valley/coast ( or Seahorse Valley East )
** main cardioid seahorse valley = Gap between the head ( period 2 component) and the body (or shoulder = main cardioid). Particularly the upper one part.
** 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
** 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.
** 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 = limiting 1/n limb of main cardioid as n tends to infinity. See demo 2 page 10 from program mandel by Wolf Jung
* Gap between the head and the body = seahorse valley
* Double-spirals on the left, seahorses on the right
* Fragment of Mandelbrot set called the elephant valley

## Mini Mandelbrot sets

Names:
* midget
* mini mandelbrot set
* baby Mandelbrots
* island = island mu-molecule = island mu-unit
* primitive component
* small copies of itself
* babies Mandelbrot sets = BMSs
Parameter rays of mini Mandelbrot sets
" a saddle-node (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 saddle-node parameter value in the corresponding parameter plane." Devaney
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 accessible from main cardioid ( M0) by a finite number of boundary crossing ?
* distorted islands

## Julia island or Embedded Julia Set or virtual Julia Sets or medallions

* "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
* "This location is called Julia island since it looks like a Julia set but it's actually inside Mandelbrot."
Medallions:
* carrot = non-spiral medallion (-1.6898799090349986e-01 1.0423707254693195e+00 3.0218142193747843e-07 ) type: long double
* cauliflower = single spiral medallion (-0.1543869 1.0308295 3.0218142193747843e-07 ) type: long double
* double spiral medallion (-0.16092059 1.03663239 0.000001 ) type: long double
* triple spiral medallion ( -1.5403777941777627e-01 1.0369221371305641e+00 6.5186720162412668e-07 ) type: long double

## hyperbolic componenets

* Components of the parameter plane

### order of hyperbolic componenets

* 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
* orbit portrait by Milnor 2000
* shrubs by M Romero et al 2004

### shape

* 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 plane

* with respect to the Mandelbrot set
** exterior
** boundary with different point types: Cremer, Siegel, parabolic, Misiurewicz, Feigenbaoum, ...
** interior
* with respect to the wakes
** inside a p/q wake
** outside any wake (???)
Parts of Mandelbrot set according to M Romera et al.:
** main cardioid
** q/p family (= q/p limb)
*** periodic part: period doubling cascade of hyperbolic components which ends at the Myrberg-Feigenbaum point
*** the Myrberg-Feigenbaum point
*** chaotic part: shrub
Not that her q/p not p/q notation is used
Components of parameter plane:
* wake / limb
* migdet / island / mu-molecule
** distorted = warped
* hyperbolic component of Mandelbrot set
Wake is a part of parameter plane between 2 parameter rays landing on the same parabolic point ( root point, bond)
Limb is a part of Mandelbrot set between 2 parameter rays landing on the same parabolic point ( root point, bond)
Strict ( math ) definition from the paper Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account by John W. Milnor
Theorem 1.2. The Wake ${\mathit {W}}_{\mathcal {P}}$. The two corresponding parameter rays ${\mathcal {R}}_{{\mathit {t}}\pm }^{\mathit {M}}$ land at a single point $\mathbf {r} _{\mathcal {P}}$ of the parameter plane. These rays, together with their landing point, cut the plane into two open subsets
* ${\mathit {W}}_{\mathcal {P}}$
* and $\mathbb {C} \setminus {\overline {{\mathit {W}}_{\mathcal {P}}}}$
with the following property: A quadratic map $f_{c}$ has a repelling orbit with portrait ${\mathcal {P}}$ if and only if $c\in {\mathit {W}}_{\mathcal {P}}$, and has a parabolic orbit with portrait ${\mathcal {P}}$ if and only if $c=\mathbf {r} _{\mathcal {P}}$.
Definitions:
* the Mandelbrot set ${\mathit {M}}$
* This open set ${\mathit {W}}_{\mathcal {P}}$ will be called the ${\mathcal {P}}$-wake in parameter space (compare Atela [A]),
* $\mathbf {r} _{\mathcal {P}}$ will be called the root point of this wake
* The intersection ${\mathit {M}}_{\mathcal {P}}={\mathit {M}}\cap {\overline {{\mathit {W}}_{\mathcal {P}}}}$ will be called the ${\mathcal {P}}$-limb of the Mandelbrot set
* The open arc $I_{S_{1}}=(t_{-},t_{+})$ consisting of
** all angles of dynamic rays ${\mathcal {R}}_{t}^{\mathcal {K}}$ which are contained in the interior of S1,
** or all angles of parameter rays ${\mathcal {R}}_{t}^{\mathcal {M}}$ which are contained in ${\mathit {W}}_{\mathcal {P}}$, will be called the characteristic arc ${\mathcal {I}}={\mathcal {I}}_{\mathcal {P}}$ for the orbit portrait ${\mathcal {P}}$
Nomenclature types (source of the names):
* mu-ency
* Mandelbrot papers and books

# How to move on parameter plane ?

* 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
* Implosion : from circle thru cauliflower to imploded cauliflower
* 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 ?

* fixed size step / adaptive size step
* minimal size of the move
What is the reason for : 2^-25 ?
Compare
* Numerical calculations and rigorous mathematics
* Reliable Computing, Interval Computations

## Types

* type of the move
** continuous
** discrete = using sequence of points
* type of the curve
** along radial curves :
*** escape route
*** external ray: From Cantor to parabolic Parameter along External Ray
*** parabolic point = root point = landing point of above external ray
*** internal ray: From Hyperbolic to Parabolic Parameters along Internal Rays
** along circular curves :
*** equipotentials
*** boundaries of hyperbolic components
*** internal circular curves
** loops
The dynamics of the polynomials of moving along curves is :
* 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

## Examples

* 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 Semi-hyperbolic Parameters along External Rays
* 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
* Cantor to Misiurewicz: along the parameter ray of angle 15/56 by Tomoki Kawahira
* Cantor to Parabolic along the parameter ray of angle 1/3: by Tomoki Kawahira
Boundary of the component
* boundary of the largest bulb of the Mandelbrot set by David Madore
Others
* morphing
* poincare_half-plane_metric_for_zoom_animation by Claude Heiland-Allen
* 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

### Escape route

* escape route 1/2

#### Escape route 0

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
contious evolution of dynamics along escape route 0 ( parabolic implosion) :
| 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:
* evolution of dynamics along internal/external ray 0
* center = superattracting
* attracting
* parabolic
* repelling
* target set
* Internal level sets
* binary decomposition
* Full tile = binary decomposition and internal level sets
Real slice of parameter plane: imag(c) = 0 :
| 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 M-set to its boundary |

#### Escape route 1/3

* evolution of dynamics along internal1/3
* 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 types

Criteria for plane classifications
* fractal formula ( function): c-plane, lambda-plane
* plane transformations
There are many different types of the parameter plane
* by function
** plain c-plane: $z=z^{2}+c$
** plain lambda plane $z=\lambda z(1-z)$ where $c={\frac {\lambda }{2}}-{\frac {\lambda ^{2}}{4}}$
* by transformations
** inverted c-plane = 1/c plane: $z=z^{2}+c^{-1}$
** exponential plane ( map)
** unrolled plain (flatten' the cardiod = unroll ) = "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 204-205 of The Science Of Fractal Images)
* c-plane
* inverted c plane = 1/c plane
* ${\frac {c^{2}}{c^{4}-0.25}}$ plane
* Unrolled main cardioid of Mandelbrot set for periods 7-13
* lambda plane
* 1/lambda plane
See also
* Mandelbrot Sets: Alternate Parameter Planes

## Parameter space types by dimensions

* 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(1-x)(x+b)/(x+d) by Valannorton
One can only show 2D slice in the multi dimensional space.

# Points

* real

## How to describe c point ?

Numerical description
* c value
** Cartesion description
*** real part
*** imaginary part
** polar description:
*** (external or internal ) angle
*** ( external or internal) radius, see stability index
Symbolic description
* set relation: Julia set interior / boundary / exterior

## How to save parameters of the point?

* parameter files: files with saved parameter values
* image files with saved parameters

## examples of important points

Examples from the parameter plane and Mandelbrot set:
* The northwest external angle is 3/8
* The north external angle is 1/4
* The northeast external angle is 1/8
* The west external angle is 1/2
* The east external angle is 0
* The southwest external angle is 5/8
* The south external angle is 3/4
* The southeast external angle is 7/8,

## Point Types

point
* pixel of parameter plane
* c parameter of complex quadratic polynomial $f_{c}(z)=z^{2}+c$
* complex number
* point coordinate

### Criteria

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
* 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), biaccessible, triaccessible, ....
** position of critical point with relation to the Julia set
* Renormalization

### Classification

There is no complete classification. The "unclassifed" parameters are uncountably infinite, as are the associated angles.

#### Simple classification

* 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 semi-hyperbolic if the critical point is non-recurrent and belongs to the Julia set
** A typical example of semi-hyperbolic parameter is a Misiurewicz point: We say a parameter ˆc is Misiurewicz if the critical point of fcˆ is a pre-periodic point.

#### partial classification of boundary points

Classification :
* Boundaries of primitive and satellite hyperbolic components:
** Parabolic (including 1/4 and primitive roots which are landing points for 2 parameter rays with rational external angles = biaccessible ).
** Siegel ( a unique parameter ray landing with irrational external angle)
** Cremer ( a unique parameter ray landing with irrational external angle)
* Boundary of M without boundaries of hyperbolic components:
** non-renormalizable (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 non-hyperbolic 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 ...

## How to choose a point from parameter plane ?

* 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)
* 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
** see also known regins in
** morphing
** interesting areas
* zoom
** autozoom
** math.stackexchange question: value-to-use-as-center-of-mandelbrot-set-zoom
** Mandelbrot Examples with parameters by David J. Eck
** examples
Is parameter tweaking an acquired art or just random chance?
Mandelbrot set : z^6+ A*z+c How do you find such intereseting examples ?

# Curves

Curves on the parameter plane
* rays
** Parameter External Ray
** internal rays
* equipotential curves
* boundary
** of whole Mandelbrot set
** of hyperbolic components

# Algorithms

* general or representation functions
** Boolean_escape_time
*** BDM/M
** Complex potential
* atom domains
* bof60
* The_Lyapunov_exponent
* True shape
* Discrete Langrangian Descriptors
* combinatorial : tuning
** wake
** principle Misiurewicz points for the wake k/r of main cardioid
** subwake, tuning and internal address
** roots, islands and Douady tuning
* Julia morphing - to sculpt shapes of Mandelbrot set parts ( zoom ) and Show Inflection

# Models

* 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
* Sharkovsky ordering

# Size or area

* mandelbrot-area by Kerry Mitchell ( 2001)
* The size of Mandelbrot bulbs by A.C. Fowler Mark J. Mcguinness

# How to to determine the ideal number of maximum iterations for an arbitrary zoom level in a Mandelbrot fractal

* a-way-to-determine-the-ideal-number-of-maximum-iterations-for-an-arbitrary-zoom
* Automatic Dwell Limit From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023.
* stackoverflow question: how-many-iterations-of-the-mandelbrot-set-for-an-accurate-picture-at-a-certain-z
* stackoverflow question: calculate-a-dynamic-iteration-value-when-zooming-into-a-mandelbrot
* Adaptive maxiter depending on the inverse square root of the magnification

# See also

* Types of Julia sets
* Exact Coordinates by  Robert P. Munafo, 2003 Sep 22.
* Rational Coordinates   by Robert P. Munafo, 2012 Apr 22.

# Rerferences

* ↑ 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 one-dimensional quadratic maps by Gerardo Pastor, Miguel Romera, Fausto Montoya Vitini
* ↑ The On-Line 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) 1987-2022
* ↑ mandelbrot-locations by woofractal
* ↑ Mandelbrot Buds and Branches by Timothy Chase
* ↑ Map of the Mandelbrot Set. Copyright © 2005-2011 Janet Parke
* ↑ the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018
* ↑ M. Romera, G. Pastor, A. B. Orue, D. Arroyo, F. Montoya, "Coupling Patterns of External Arguments in the Multiple-Spiral Medallions of the Mandelbrot Set", Discrete Dynamics in Nature and Society, vol. 2009, Article ID 135637, 14 pages, 2009. https://doi.org/10.1155/2009/135637
* ↑ Parameter rays of mini mandelbrot sets
* ↑ Devaney In Global Analysis of Dynamical Systems, ed.: H. Broer, B. Krauskopf, G. Vegter. IOP Publishing (2001), 329-338 or Homoclinic Points in Complex Dynamical Systems
* ↑ embedded julia set from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2017.
* ↑ 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) 1987-2020.
* ↑ fractalforums.com : how-distorted-can-a-minibrot-be
* ↑ distribution From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020
* ↑ Intersection_(set_theory) in wikipedia
* ↑ Mu-Ency - The Encyclopedia of the Mandelbrot Set by R Munafo
* ↑ fractalforums.org : julia-sets-true-shape-and-escape-time
* ↑ fractalforums.org : julia-sets-true-shape-and-escape-time
* ↑ From Cantor to Semi-hyperbolic Parameter along External Rays by Yi-Chiuan Chen and Tomoki Kawahira
* ↑ From Hyperbolic to Parabolic Parameters along Internal Rays by Yi-Chiuan Chen and Tomoki Kawahira
* ↑ math.stackexchange question: parameter-plane-dynamics-of-fixed-points-and-their-preimages-for-standard-quadra
* ↑ Reading escaping trees from Hubbard trees in Sn by Matthieu Arfeux
* ↑ From Cantor to Semi-hyperbolic Parameter along External Rays March 2018Transactions of the American Mathematical Society 372(11) DOI: 10.1090/tran/7839 Yi-Chiuan ChenTomoki KawahiraTomoki Kawahira
* ↑ Alternate Parameter Planes by David E. Joyce
* ↑ exponentialmap by Robert Munafo
* ↑ Twisted Mandelbrot Sets by Eric C. Hill
* ↑ On quasi-conformal (in-) compatibility of satellite copies of the Mandelbrot set: I by Luna Lomonaco, Carsten Lunde Petersen
* ↑ mu-ency : exponential map by R Munafo
* ↑ Exponential mapping and OpenMP by Claude Heiland-Allen
* ↑ exponential_mapping_with_kalles_fraktaler by Claude Heiland-Allen
* ↑ Linas Vepstas : Self Similar?
* ↑ the flattened cardioid of a Mandelbrot by Tom Rathborne
* ↑ Stretching cusps by Claude Heiland-Allen
* ↑ Totally real points in the Mandelbrot Set by Xavier Buff, Sarah Koch 2022
* ↑ North by  Robert P. Munafo, 2010 Sep 20. From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2022.
* ↑ Northeast by Robert P. Munafo, 2010 Sep 20. From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2022.
* ↑ From Cantor to Semi-hyperbolic Parameter along External Rays March 2018Transactions of the American Mathematical Society 372(11) DOI: 10.1090/tran/7839
* ↑ stackexchange : classification-of-points-in-the-mandelbrot-set
* ↑ fractalforums : parameter-adjustment-art-or-luck ?
* ↑ interesting c points by Owen Maresh
* ↑ Visual Guide To Patterns In The Mandelbrot Set by Miqel
* ↑ fractalforums : deep-zooming-to-interesting-areas
* ↑ fractalforums.org : parameter-adjustment-art-or-luck
* ↑ fractalforums.org: julia-sets-true-shape-and-escape-time