Fractals/Iterations in the complex plane/Parameter plane

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Parts of parameter plane[edit]

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 Myrberg-Feigenbaum 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 :

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 ( c-plane )
  • inverted c-plane = 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 204-205 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]

point c description[edit]

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

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
  • 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:
    • 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[edit]


Models[edit]

See also[edit]

Rerferences[edit]

  1. SHRUBS IN THE MANDELBROT SET ORDERING by M Romero, G Pastor, G Alvarez, F Montoya
  2. interesting c points by Owen Maresh
  3. Visual Guide To Patterns In The Mandelbrot Set by Miqel
  4. fractalforums : deep-zooming-to-interesting-areas
  5. Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
  6. Alternate Parameter Planes by David E. Joyce
  7. exponentialmap by Robert Munafo
  8. mu-ency : exponential map by R Munafo
  9. Exponential mapping and OpenMP by Claude Heiland-Allen
  10. Linas Vepstas : Self Similar?
  11. the flattened cardioid of a Mandelbrot by Tom Rathborne
  12. Stretching cusps by Claude Heiland-Allen
  13. Twisted Mandelbrot Sets by Eric C. Hill
  14. FRONTIERS IN COMPLEX DYNAMICS by CURTIS T. MCMULLEN
  15. stackexchange : classification-of-points-in-the-mandelbrot-set
  16. fractalforums : tricky-mandelbrot-problem