# Fractals/Iterations in the complex plane/Parameter plane

## Contents

# 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:

- 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
^{[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:
- 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]

# Rerferences[edit]

- ↑ 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