# 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 ?

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

# 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 [2]

There are many different types of the parameter plane[3] [4]

• plain ( c-plane )
• inverted c-plane = 1/c plane
• exponential plane ( map) [5][6]
• unrolled plain (flatten' the cardiod = unroll ) [7][8] = "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)[9]
• transformations [10]

# Point Types

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

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

A partial classification of boundary points would be :[11]

• 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 ...

# Rerferences

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