This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
Most of what you could want to know is in Branner's article in
"Chaos and Fractals: The Mathematics Behind the Computer Graphics".
Note that the Mandelbrot set in general is "not" strictly
self-similar; the tiny copies of the Mandelbrot set are all slightly
different, mainly because of the thin threads connecting them to the
main body of the Mandelbrot set. However, the Mandelbrot set is
quasi-self-similar. However, the Mandelbrot set is self-similar under
magnification in neighborhoods of Misiurewicz points (e.g.
-.1011 + .9563i). The Mandelbrot set is conjectured to be self-similar
around generalized Feigenbaum points (e.g. -1.401155 or
-.1528 + 1.0397i), in the sense of converging to a limit set.
1. T. Lei, Similarity between the Mandelbrot set and Julia Sets,
"Communications in Mathematical Physics" 134 (1990), pp. 587-617.
2. J. Milnor, Self-Similarity and Hairiness in the Mandelbrot Set, in
"Computers in Geometry and Topology", M. Tangora (editor), Dekker,
New York, pp. 211-257.
The "external angles" of the Mandelbrot set (see Douady and Hubbard or
brief sketch in "Beauty of Fractals") induce a Fibonacci partition
The boundary of the Mandelbrot set and the Julia set of a generic c in
M have Hausdorff dimension 2 and have topological dimension 1. The
proof is based on the study of the bifurcation of parabolic periodic
points. (Since the boundary has empty interior, the topological
dimension is less than 2, and thus is 1.)
1. M. Shishikura, The Hausdorff Dimension of the Boundary of the
Mandelbrot Set and Julia Sets, The paper is available from
anonymous ftp: ftp://math.sunysb.edu/preprints/ims91-7.ps.Z