I am trying to understand the classification of points in the Mandelbrot set. There are an infinite number of baby Mandelbrots, each associated with a defined set of landing rays. There are the pre periodic Misiurewicz branching points, which also have landing rays. What are the rest of the points in the Mandelbrot set called? Are they called chaotic? Also, is this set of remaining uncountably infinite? What is known about landing rays for these remaining points?
Criteria for classification :
- area / Lebesgue measure
- set properities ( interior, exterior, boundary )
- arithmetic properties of internal angle ( rotational number)
- landing of external rays ( for boundary points ) : biaccesible, triaccesible
- external/ internal angle and complex potential
New answer made by Wolf Jung :
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 or non-hyperbolic ) components:
- Parabolic (including 1/4 and primitive roots).
- Siegel ( there is a unique parameter ray landing )
- Cremer ( there is a unique parameter ray landing )
- Boundary of M without boundaries of ( hyperbolic or non-hyperbolic ) components:
- non-renormalizable (Misiurewicz and other).
- finitely renormalizable (Misiurewicz and other).
- infinitely renormalizble (Feigenbaum and other).
non hyperbolic components : we believe they do not exist but we cannot prove it ; infinitely renormalizable
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 ...