Fractals/Iterations in the complex plane/Parameter plane

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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 example (parabolic point ) choose hyperbolic component ( period , number) and internal angle (= rotation number) then compute c parameter.

How to move on parameter plane ?[edit]

  • along radial curves :
    • external rays
    • parabolic points
    • internal rays
  • along circular curves :
    • equipotentials
    • boundaries of hyperbolic components
    • internal circular curves

Plane types[edit]

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 [1]

There are many different types of the parameter plane[2] [3]

  • plain ( c-plane )
  • inverted c-plane = 1/c plane
  • exponential plane ( map) [4][5]
  • unrolled plain (flatten' the cardiod = unroll ) [6][7] = "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)[8]
  • transformations [9]

Point Types[edit]

Criteria for classification of parameter plane points :

  • arithmetic properties of internal angle ( rotational number)
  • landing of external rays ( for boundary points ) : biaccesible
  • Renormalization
  • set properties ( interior of set and set )

Simple classification of parameter plane points :

  • exterior of Mandelbrot set
  • Mandelbrot set
    • boundary of Mandelbrot set
    • interior of Mandelbrot set

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

A partial classification of boundary points would be :[10]

  • 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).
    • 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 ...

See also[edit]


  1. Lasse Rempe, Dierk Schleicher : Bifurcation Loci of Exponential Maps and Quadratic Polynomials: Local Connectivity, Triviality of Fibers, and Density of Hyperbolicity
  2. Alternate Parameter Planes by David E. Joyce
  3. exponentialmap by Robert Munafo
  4. mu-ency : exponential map by R Munafo
  5. Exponential mapping and OpenMP by Claude Heiland-Allen
  6. Linas Vepstas : Self Similar?
  7. the flattened cardioid of a Mandelbrot by Tom Rathborne
  8. Stretching cusps by Claude Heiland-Allen
  9. Twisted Mandelbrot Sets by Eric C. Hill
  10. stackexchange : classification-of-points-in-the-mandelbrot-set