Fractals/Iterations in the complex plane/Parameter plane

Parts of parameter plane

• with respect to the Mandelbrot set
  • exterior
  • boundary
  • 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)
    • 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

How to choose a point from parameter plane ?

• 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
  • see also known regins in
  • morphing
  • intresting areas
• zoom
  • autozoom

How to move on parameter plane ?

Types

• type of the move
  • continous
  • 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 end at the parabolic point
  • along circular curves :
    • equipotentials
    • boundaries of hyperbolic components
    • internal circular curves

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

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 types

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

There are many different types of the parameter plane

• plain ( c-plane )
• inverted c-plane = 1/c plane
• lambda plane
• 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)
• transformations
• 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. "

• 

  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

Transformations

• 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."

point c description

• c value
  • Cartesion description
    • real part
    • imaginary part
  • polar description:
    • (external or internal ) angle
    • ( external or internal) radius

Point Types

point =pixel of parameter plane = c parameter

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 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

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
    • centers,
    • other internal points ( points of internal rays )

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 = biaccesible ).
  • 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 critcal point may be rcurrent or not, the number of branches at branch points may be bounded or not ...

Algorithms

• 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
• zoom
  • Perturbation_theory
• 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

See also

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