# Fractals/Iterations in the complex plane/Parameter plane

# StructureEdit

## real sliceEdit

" each period-doubling cascade superstable orbit ... generate the corresponding chaotic band and the Misiurewicz point that separates the chaotic bands and "^{[1]}

where:

- B is a chaotic region from -2 to MF
- = Feigenbaum point = MF = the Myrberg-Feigenbaum
- A is a periodic region : from MF to 1/4

### chaotic regionEdit

where:

- is chaotic band

```
```

See also

### periodic regionEdit

- 2^n = the powers of 2 in decreasing order, ending with 2^0 = 1."
^{[2]}

## Mini Mandelbrot setsEdit

Name:

- midget
- mini mandelbrot set
- baby Mandelbrots
- island = island mu-molecule = island mu-unit
^{[3]} - primitive component

Parameter rays of mini Mandelbrot sets ^{[4]}

" 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 ^{[5]}

" Warped midgets in the Mandelbrot set have been measured, using an algorithm that allows the positions of the head, and cardioid atoms (north and south) of any midget to be found, once one has placed the cursor on the computer terminal somewhere inside any midget. We describe two distortions of midgets: linear distortions and angular distortions. When the north and south angles are plotted in the north/south angle plane, families of points are formed. The angle and distance measures of warped midgets from the Sea Horse Valley of the Mandelbrot set and from other sea horse valleys of midgets, whether on the Spike or on tendrils above atoms, all fall closely together in one part of the north/south plane. Measures of warped midgets from tendrils above the major atoms on the surface of the Cardioid fall closely together in another part of the north/south plane. This different way of looking at the Mandelbrot set offers an interesting way of studying the distortions of midgets." A. G. Davis Philip, Michael Frame, Adam Robucci: Warped midgets in the Mandelbrot set. Computers & Graphics 18(2): 239-248 (1994)

Distortion:

- Baby Mandelbrot Sets are Not Exact Self Similar. A baby Mandelbrot sets are homeomorphic to the original one
- fractalforums : how-distorted-can-a-minibrot-be/
- math.stackexchange question: mini-mandelbrots-are-they-exact-copies ?
- math.stackexchange question: why-does-the-mandelbrot-set-contain-slightly-deformed-copies-of-itself

## 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
^{[6]} - "This location is called Julia island since it looks like a Julia set but it's actually inside Mandelbrot."
^{[7]}

# 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
^{[8]} - orbit portrait by Milnor 2000
- shrubs by M Romero et al 2004
^{[9]}

## 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.:^{[10]}

- main cardioid
- q/p family (= q/p limb)
- period doubling cascade of hyperbolic components which ends at the Myrberg-Feigenbaum point
- shrub

Not that her q/p not p/q notation is used

ComponentsL

- wake / limb
- migdet / island / mu-molecule
^{[11]}- distorted
^{[12]}= warped^{[13]}

- distorted
- hyperbolic component of Mandelbrot set

### namesEdit

Source

- mu-ency
^{[14]} - 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)
^{[15]}

- 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
^{[16]} - see also known regins in
^{[17]} - morphing
- interesting areas
^{[18]}

- zoom

# How to move on parameter plane ?Edit

## TypesEdit

- type of the move
- continuous
- discrete = using sequence of points

- type of the curve
- along radial curves :
- 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
^{[19]}

- along radial curves :

The dynamics of the polynomials of moving along curves is : ^{[20]}

- 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

Examples :

- 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

// glsl code by iq from https://www.shadertoy.com/view/Mss3R8 float ltime = 0.5-0.5*cos(time*0.12); vec2 c = vec2( -0.745, 0.186 ) - 0.045*zoom*(1.0-ltime);

// 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));

# Plane typesEdit

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
^{[21]}

There are many different types of the parameter plane^{[22]}^{[23]}

- plain ( c-plane ):
- inverted c-plane = 1/c plane:
- lambda plane where
- exponential plane ( map)
^{[24]}^{[25]}^{[26]} - unrolled plain (flatten' the cardiod = unroll )
^{[27]}^{[28]}= "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)^{[29]} - transformations
^{[30]} - log : "To illustrate the complexity of the boundary of the Mandelbrot set, Figure 8 renders the image of dM under the transformation log(z - c) for a certain c e dM ? Note the cusp on the main cardioid in the upper right; looking to the left in the figure corresponds to

zooming in towards the point c. (Namely, c = -0.39054087... - 0.58678790i... the point on the boundary of the main cardioid corresponding to the golden mean Siegel disk.). Note the cusp on the main cardioid in the upper right; looking to the left in the figure corresponds to
zooming in towards the point c. "^{[31]}

- "Legendary side scrolling fractal zoom. 1 Month + (Interpolator+Video Editor) = Log(z). This means logarithmic projection for this location, that gives this interesting side-scrolling plane ^^)"
^{[32]} - " There are no program that can render this fractal on log(Z) plane. But you can make it in Ultra Fractal or in similar software with programmable distributive. Formula is:C = exp(D), for D - is your zoomable coordinates" SeryZone X

- "Legendary side scrolling fractal zoom. 1 Month + (Interpolator+Video Editor) = Log(z). This means logarithmic projection for this location, that gives this interesting side-scrolling plane ^^)"

Description by NIKOLA UBAVIĆ:

- Inverted c-plane : "Geometrically, the ... relationship between the parameters represents the composition of the inversion with respect to the unit circle centered at zero, and the conjugation (axial symmetry with respect to the real line). Due to this connection, the cardioid from the boundary of the Mandelbrot set in the "standard" parameterization corresponds to the tear-shaped curve in the alpha parameterization."
^{[33]} - "if translation is performed for before the inversion 1/4, then the cardioid is imaged in a parabola"
- "By inverting a complex plane around a unit circle with center at zero, one of these circles remains invariant, while the other image is inside it."
- "If ... translation is performed before the inversion1, then the two circles are mapped into two parallel lines. In this way the second of the next two figures was obtained."

## TransformationsEdit

- description
- examples
- Mandelbrot set projected on a shrinking Riemann-sphere by Arneauxtje
- Mandelbrot's Elephant Valley (Short Version) Timothy Chase
- Mandelbrot Buds and Branches Timothy Chase Timothy Chase
- Mandelbrot Zoom on a Sphere video by craftvid : "This a 300-trillion time zoom-in on the Mandelbrot set. The images are set on a Spherical "mobius" projection, meant to be wrapped onto a spherical surface. The image zooms in on the front center of the Sphere, while fading away on the back of the sphere."
- Cat Eye : Inverted Mandelbrot Set by denis archambaud
- Inverse & Potens Mandelbrot Set by Jens-Peter Christensen
- Z² + Sin(Cˉᵐ+phase) by Jens-Peter Christensen

# point c descriptionEdit

- c value
- Cartesion description
- real part
- imaginary part

- polar description:
- (external or internal ) angle
- ( external or internal) radius

- Cartesion description

# 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 properities
- 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
- centers,
- other internal points ( points of internal rays )

### partial classification of boundary pointsEdit

Classification :^{[34]}

- Boundaries of primitive and satellite hyperbolic components:
- 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 ...

# 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 non-attracting Basins and Petals by Chris King
- atom domains
- average distance between random points
^{[35]}

- 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: mandelbrot-image-in-every-language
- codegolf SE question: generate-a-mandelbrot-fractal ( ASCI)
- rosetta code : Mandelbrot_set
- stackoverflow question: code-golf-the-mandelbrot-set
- stackoverflow questions tagged mandelbrot
- benchmark

# ModelsEdit

Structure of the Mandelbrot set

# See alsoEdit

# RerferencesEdit

- ↑ [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
- ↑ the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018
- ↑ 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
- ↑ Mu-Ency - The Encyclopedia of the Mandelbrot Set by R Munafo
- ↑ 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
- ↑ 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
- ↑ Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
- ↑ Alternate Parameter Planes by David E. Joyce
- ↑ exponentialmap by Robert Munafo
- ↑ 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
- ↑ Twisted Mandelbrot Sets by Eric C. Hill
- ↑ FRONTIERS IN COMPLEX DYNAMICS by CURTIS T. MCMULLEN
- ↑ youtube video : Mandelbrot deep zoom to 2^142 or 5.5*10^42. Log(z) by SeryZone X
- ↑ Julie's and Mandelbrot's set by NIKOLA UBAVIĆ
- ↑ stackexchange : classification-of-points-in-the-mandelbrot-set
- ↑ fractalforums : tricky-mandelbrot-problem