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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?

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+1 and added to favorites :) –  mick Jan 29 '13 at 22:22
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2 Answers

up vote 6 down vote accepted

boundary points of Mandelbrot set ( c-points ) :

  1. Siegel points

  2. parabolic points = root points of hyperbolic components = biaccesible points ( landing points of 2 rays )

  3. Misiurewicz points

  4. Chaotic points

  5. Cremer points

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Nice answer. +1 –  mick Jan 29 '13 at 22:23
    
I think this answer is incomplete. What about the tip of the Mandelbrot, where, after an infinite number of bifurcations, we have more or less infinite chaos? I would guess that this point is on the boundary of the m-set. Also, it would seem that a little bit of elaboration on which points are known to have landing rays; which includes the rationally indifferent parabolic points. Cremer points are irrationally indifferent orbits on Julia sets; does this apply to the mandelbrot set. It is not known whether Cremer points have landing rays, what about Siegel disc points on the mandelbrot set? –  Sheldon L Aug 28 '13 at 10:06
    
Hey Adam, if you get this message, find some way to update your answer, and I'll give you the "Plus" back. Maybe I'm wrong about the tip of the mandelbrot. "My vote is now locked in until you edit the answer". Apologies in advance .... –  Sheldon L Aug 28 '13 at 19:44
    
Thx for pointing it out. I have added one item : Misiurewicz points. Tip of M set, you mean c=-2 ? It is a Misiurewicz point M_2,1. –  Adam Aug 28 '13 at 19:54
    
I wasn't thinking c=2; but thanks for adding Misiurewicz points! That was was also missing. I was thinking about the limit of Feigenbaum infinite bifurcation, $C\approx-1.401155189092051$. Is there a special name for the Feigenbaum limit of infinite bifurcation? Also, I would add the term parabolic cusp to the answer as well. I think this is a special parabolic case, that occurs only at the cusp. Is parabolic cusp the correct term for the butt of the Main Cardioid and also similarly for all mini-Mandelbrot cardiods, each of which has one parabolic cusp point? –  Sheldon L Aug 28 '13 at 22:09
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Criteria for classification :

  • area / Lebesgue measure
  • set properities ( interior, exterior, boundary )
  • renormalization
  • 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).

Notes :

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 ...

non-renormalizable parameters : those that do not belong to any small copies of the Mandelbrot set

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What is the difference between a Misiurewicz non-renormalizeable as compared to a Misiurewicz finitely renormalizeable point? –  Sheldon L Sep 29 '13 at 3:42
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