Yeah, that must be it. Thanks for the link to that interview. *I put as a conjecture, and the conjecture is that the first set which I was considering - call it M0 - if you add it to all its limit points, make it the limit points plus the set, you obtain the set M which is obtained by connectedness of the Julia set. This conjecture looked extremely simple. It can be explained to a good high school student, because there was no complicated concept involved. The concept of a limit cycle is straightforward. The concept of connectedness is straightforward and intuitive. The relation between the two was postulated by a very simple identity, that M equals M0 plus its limit points. Well, believe it or not, this conjecture is still open and what is so striking about the study of iteration of Z squared plus C is that the first serious difficulty I encountered in its study has remained, after now eighteen, nineteen years of study, totally baffling.* - Benoit Mandelbrot (May 1998) Even simpler than the lemniscate thing, and clearly an easy thing to assume given how Mandelbrot defines and discusses the separator set (page 183 of his 1982 book[1]). Odd, I thought Fatou had proven that one, but I was wrong (-: [1] Benoit B. Mandelbrot. The Fractal Geometry of Nature. New York: W. H. Freeman and Company, 1983. ISBN 0-7167-1186-9. On Thu, Jul 28, 2011 at 10:28, Tom Karzes <karzes@sonic.net> wrote:
I believe this is what Mandelbrot was referring to. Here's another interview with him in which he talks very specifically about it:
http://www.webofstories.com/play/10523
Tom
Dan Asimov writes:
Suddenly two definitions of the Mandelbrot set come to mind, the first one I learned (I), and the much more common one (II):
For any c in C, define f_c(z) as z^2 + c.
I. The set of c in C for which the Julia set of f_c is connected. (See < http://en.wikipedia.org/wiki/Julia_set >.)
II. the set of c in C for which the orbit of 0, under (forward) iteration of f_c, is bounded.
I've never seen a proof that these definitions are equivalent, though I haven't looked very hard, either. (In II, it seems to me that considering the orbits of 0 rather than of any other point is somewhat arbitrary.)
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