SIMILARITY BETWEEN THE MANDELBROT SET AND JULIA SETS

The Mandelbrot set M is "self-similar" about any Misiurewicz point c in the sense that if we examine a neighborhood of c in M with a very powerful microscope, and then increase the magnification by a carefully chosen factor, the picture will be unchanged except for a rotation. The corresponding Julia set J(c) is also "self-similar" in the same sense, with the same magnification factor. Moreover, the two sets M and J(c) are "similar" in the sense that if we use a very powerful microscope to look at M and J(c), both focused at c, the structures we see look like very much the same.