Fractals/Iterations in the complex plane/Parameter plane
Contents
Parts of parameter plane[edit]
 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.:^{[1]}

 main cardioid
 q/p family (= q/p limb)
 period doubling cascade of hyperbolic components which ends at the MyrbergFeigenbaum point
 shrub
Not that her q/p not p/q notation is used
How to choose a point from parameter plane ?[edit]
 clicking on parameter points and see what you have ( random choose)
 computing a point with known properties.
 For (parabolic point ) choose hyperbolic component ( period , number) and internal angle (= rotation number) then compute c parameter.
 Misiurewicz points ^{[2]}
 see also known regins in ^{[3]}
 morphing
 intresting areas^{[4]}
 zoom
How to move on parameter plane ?[edit]
 along radial curves :
 external rays
 parabolic points
 internal rays
 along circular curves :
 equipotentials
 boundaries of hyperbolic components
 continously
 using only sequences of points and skipping others on the boundary of hyperbolic component
 internal circular curves
Examples :
 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
Plane types[edit]
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 ^{[5]}
There are many different types of the parameter plane^{[6]} ^{[7]}
 plain ( cplane )
 inverted cplane = 1/c plane
 lambda plane
 exponential plane ( map) ^{[8]}^{[9]}
 unrolled plain (flatten' the cardiod = unroll ) ^{[10]}^{[11]} = "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)^{[12]}
 transformations ^{[13]}
 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. "^{[14]}
Transformations[edit]
 description
 examples
 Mandelbrot set projected on a shrinking Riemannsphere 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 300trillion time zoomin 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."
point c description[edit]
 c value
 Cartesion description
 real part
 imaginary part
 polar description:
 (external or internal ) angle
 ( external or internal) radius
 Cartesion description
Point Types[edit]
point =pixel of parameter plane = c parameter
Criteria[edit]
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 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
Classification[edit]
There is no complete classification. The "unclassifed" parameters are uncountably infinite, as are the associated angles.
Simple classification[edit]
 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 points[edit]
Classification :^{[15]}
 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 critcal point may be rcurrent or not, the number of branches at branch points may be bounded or not ...
Algorithms[edit]
 General
 Escape time
 DEM/M
 Discrete Velocity of nonattracting Basins and Petals by Chris King
 atom domains
 average distance between random points^{[16]}
 zoom
 combinatorial : tuning
Models[edit]
See also[edit]
 Types of Julia sets
 Exact Coordinates by Robert P. Munafo, 2003 Sep 22.
 Rational Coordinates by Robert P. Munafo, 2012 Apr 22.
Rerferences[edit]
 ↑ SHRUBS IN THE MANDELBROT SET ORDERING by M Romero, G Pastor, G Alvarez, F Montoya
 ↑ interesting c points by Owen Maresh
 ↑ Visual Guide To Patterns In The Mandelbrot Set by Miqel
 ↑ fractalforums : deepzoomingtointerestingareas
 ↑ 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
 ↑ 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
 ↑ FRONTIERS IN COMPLEX DYNAMICS by CURTIS T. MCMULLEN
 ↑ stackexchange : classificationofpointsinthemandelbrotset
 ↑ fractalforums : trickymandelbrotproblem