Fractals/Iterations in the complex plane/Parameter plane
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 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 ...