It is often stated that fractals, such as the Mandelbrot set M, are self-similar, although I've never heard of any functions to formally model this perspective.

I'm curious to learn about any functions f that map M to itself in nontrivial ways. By "nice" in the title, I mean to exclude trivial or very artificial functions that fail to provide any insight into the structure of M (so "nice" is being used informally).


The Mandelbrot set is not self-similar in the sense that you mean -- that applies more to fractals like the Sierpinski triangle, the Koch curve, and so forth. There are several subsets of the Mandelbrot set that are approximately similar to the Mandelbrot set, as shown in this zoom sequence from Wikipedia:

Quasi-self-similarity of the Mandelbrot set

The Mandelbrot set is also infinitesimally self-similar at any Misiurewicz point, in the sense that zooming in at such a point reveals local structure that seems to repeat the farther you zoom in. (In fact, the local structure is converging to the local structure of the corresponding Julia set.)

As a general rule, not all fractals are self-similar, though self-similarity is a common property of simple fractal shapes. There is not a single widely-accepted definition of fractal within mathematics, though it is generally agreed that any shape with non-integer Hausdorff dimension is a fractal. The Mandelbrot set has Hausdorff dimension 2, but it is still considered a fractal because of its complicated local structure and approximate self-similarities.

If you are looking for some insight into the structure of $M$, here's a good way to proceed:

  1. Learn about the main cardioid and period bulbs, and possibly the antennas as well. (This article by Bod Devaney might be a good place to start.)

  2. Learn something about postcritically finite quadratics, and their relationship to hyperbolic components and Misiurewicz points.

  3. Learn about external rays and Bötcher coordinates on the complement.

  4. Learn about tuning and renormalization.

  • $\begingroup$ Thanks for the interesting links, @Jim. Is there any particular reason to think such functions M->M don't exist? $\endgroup$ – Tyler Jun 20 '11 at 22:57
  • $\begingroup$ I think this is what tuning and renormalization are about. The resulting functions are not similarity transformations, but they do explain the quasi-self-similarity of the Mandelbrot set. $\endgroup$ – Jim Belk Jun 20 '11 at 23:37

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