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I recently learned about the Mandelbrot set for the first time from a presentation by some undergraduates in honor of Mandelbrot's death. The presentation was short and by non-experts so I left with a few questions.

When I heard about the fundamental dichotomy, it seemed odd to me that the attractive basin at infinity was a distinguished point, in that filled Julia sets are the complement of the attractive basin of infinite. But when I naively expected the set of bounded points to be the attractive basin of the origin I was caught by the lack of duality--the filled Julia set is either simply connected or a Cantor set, and the complement of a simply connected set is simply connected, while I have only the slightest idea what the complement of a Cantor set looks like; it certainly isn't simply connected.

Talking the subject over with a few graduate students I realized that there should be at least countably many bounded attractive basins or loops for any curve, and it seems unlikely that in the case that the basin of infinity is the complement of a Cantor set that the other attractive basins (including the loops of the roots of $f ^{n}(x) - x\ $, where $f(x) = z^2+c$ ) are also complements of Cantor sets.

When $c=0$ the picture is very nice and symmetric, with two simply connected attractive basins, countably many finite loops at the $\pi$-rational angles around the unit circle, and uncountably many infinite loops. I'm curious what the picture looks like for other $c$.

So my questions are:

What distinguishes the point at infinity from the other points here?

What do the loops look like?

Are there other attractive basins? For which $c$? What do they look like?

Or what book(s) or paper(s) could I look through to get satisfying answers to these questions?

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perhaps this question would benefit from a "complex dynamics" tag, or something similar – Yemon Choi Nov 1 '10 at 4:14
Here is a naïve observation. The asymmetry is related to the fact that the family of functions of the form $z \mapsto z^2 + c$ is not invariant under any Möbius transformations. In particular, if we conjugate by a map that switches zero and infinity, e.g., $w = 1/z$, we end up with $w \mapsto w^2/(1+cw^2)$, which has qualitatively different behavior (as you would expect). – S. Carnahan Nov 1 '10 at 4:54
Your questions appear quite vague. The math.stackexchange site might be more appropriate. But perhaps more appropriate would be to pick up a book like Devaney's "An Introduction To Chaotic Dynamical Systems." What is your background? Devaney has some undergraduate oriented textbooks as well. – Ryan Budney Nov 1 '10 at 5:17
You might want to look at for a discussion of the Mandelbrot and Julia sets. Also, $\infty$ is a superattracting fixed point for a polynomial (this is related to Scott's observation that polynomials aren't invariant under Mobius transformations), which is not the case for the other points. – David Roberts Nov 1 '10 at 5:31
Loop -> orbit, I think... – Sam Nead Nov 1 '10 at 11:22

It seems you are basically interested in an introduction to complex dynamical systems. The books by Beardon, Milnor and Steinmetz all give good introductions.

Regarding your specific questions:

a) The point at infinity is a superattracting fixed point, but more importantly its immediate basin of attraction - that is, the component of the basin containing the fixed point itself - is completely invariant (invariant under forward and backwards iteration). This is the case for all polynomials (of degree at least two), and is one of the reasons that studying polynomials is easier than studying general rational maps (where e.g. the Julia set - where the dynamics is chaotic - may in fact be the whole Riemann sphere). The basin of infinity supports foliations into "external rays" and "equipotentials", and this allows one to study the Julia set. This idea was introduced by Douady and Hubbard, and is the basis of the famous "Yoccoz puzzle".

b) There are all kinds of possible orbits in the Julia set. This isn't the place to discuss them. Stable orbits of polynomials may converge to a (super)-attracting fixed point, to a parabolic fixed point (where the multiplier is a root of unity), or belong to a rotation domain (a simply connected domain on which the dynamics is conjugate to a rotation).

c) See above. There may be at most one attracting cycle for a quadratic polynomial (apart from infinity), because there is at most one critical point. (This is a theorem of Fatou.) Conjecturally, systems with an attracting cycle are dense in the Mandelbrot set; this is perhaps the most celebrated conjecture in one-dimensional dynamics at the moment.

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