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Ie, is there a way to probe it for regions of depth that involves a function, the domain of which is the Mandelbrot set itself, or a part of that set?

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To be honest, I have no idea what you are asking about? Can you explain a bit more what you want to know? – Mariano Suárez-Alvarez Jul 19 '10 at 6:02
The Mandelbrot set is very deep in some places and very shallow in others. Some people have managed to produce computer graphics that continuously zoom on one bit of the set. Without recourse to some function telling them 'where to put the crosshairs,' so to speak, how could they produce such graphics? Poking around, I suppose -- a 'brute force approach.' But there must be a mathematical way. – ardentMirage Jul 19 '10 at 6:32
Do "deep" and "shallow" here have rigorous definitions? – Kevin H. Lin Jul 19 '10 at 19:42
up vote 7 down vote accepted

It is a little bit difficult to answer the question as posed, because there is a question as to what you mean by "depth".

One of the previous answers mentions Misiurewicz points - parameters where the critical orbit is pre-periodic. Examples are the "branch points" and "tips" in the Mandelbrot set. However, these are not likely to be good candidates for what you are looking for. Indeed, rescalings of the Mandelbrot set near such a parameter will converge, with the same limit as corresponding rescalings in the dynamical plane, by a theorem of Tan Lei. This does not seem like what you are looking for.

If you pick a "Feigenbaum point" (an infinitely renormalizable parameter of bounded type, such as the famous Feigenbaum value which is the limit of the period-2 cascade of bifurcations), then Milnor's hairiness conjecture, proved by Lyubich, states that rescalings of the Mandelbrot set converge to the entire complex plane. So there is certainly a lot of thickness near such a point, although again this may not be what you are looking for. It may also prove computationally intensive to produce accurate pictures near such points, because the usual algorithms will end up doing the maximum number of iterations for almost all points in the picture.

In this case you are left with either non-renormalizable parameters (those that do not belong to any small copies of the Mandelbrot set) or infinitely renormalizable parameters with interesting behaviour.

An example of the formal is given by the so-called "Fibonacci parameter"; see "Parameter scaling for the Fibonacci point" by Wenstrom for information (and a picture) about the parameter space structure near this point. Rodrigo Perez investigated the structure of parameter space near such pieces in much more detail.

To imagine creating a zoom sequence for infinitely renormalizable parameters, imagine the following: Begin with a Misiurewicz parameter c_0. Then pick a little Mandelbrot copy M_1 near c_0, and choose a Misiurewicz point c_1 in this little copy. Pick a little Mandelbrot copy M_2 very close to c_1, and again a Misiurewicz point c_2 in there, and so on. Zooming in on the limit parameter is essentially the same as zooming in to the Misiurewicz parameter c_0, then changing track and zooming into the little Mandelbrot copy M_1, centering on c_1, and so on. Of course instead of using Misiurewicz points, you could at various types use any other types of points that are on the boundary of the Mandelbrot set (boundary points of interior components, Feigenbaum parameters, etc).

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It seems to me that what you want is simply a point on the boundary of the Mandelbrot set: every neighborhood of such a point contains points both inside and outside the set. Since the boundary of the Mandelbrot set is (AFAIK) nowhere smooth, this ensures that you'll keep seeing interesting features no matter how much you zoom in on such a point.

Since the boundary of the Mandelbrot set has Hausdorff dimension 2, there should be plenty of such points to find. I don't really know much about finding explicit coordinates for such points, but a few minutes of searching suggest that the Wikipedia article on Misiurewicz points might be a good place to start.

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Chaos and Fractals: New Frontiers of Science, by Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe Chapter 14

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