Fractals/Iterations in the complex plane/Parameter plane
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 example (parabolic point ) choose hyperbolic component ( period , number) and internal angle (= rotation number) then compute c parameter.
How to move on parameter plane ?
- along radial curves :
- external rays
- parabolic points
- internal rays
- along circular curves :
- boundaries of hyperbolic components
- internal circular curves
The phase space of a quadratic map is called its parameter plane. Here:
is constant and 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 
- plain ( c-plane )
- inverted c-plane = 1/c 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 
Criteria for classification of parameter plane points :
- arithmetic properties of internal angle ( rotational number)
- landing of external rays ( for boundary points ) : biaccesible
- 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 :
- Boundaries of primitive and satellite hyperbolic components:
- 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 ...
- 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
- 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
- stackexchange : classification-of-points-in-the-mandelbrot-set