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Author Topic: Is the mandelbrot set connected?  (Read 3128 times)
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aluminumstudios
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« on: March 01, 2010, 11:21:47 AM »

Forgive me if this has been discussed a lot before or if this isn't the right board for it.  I did a quick search but couldn't find a clear answer.

My question is are all of the mini-mandelbrots in the mandelbrot-set connected?  A few places on-line talk about conjecture that it is connected and proofs that a finite number might be connected, but what I found wasn't very clear to me.

I'm just curious about it.  Any insight any could give me into if all of the mandelbrot set is connected would be appreciated.

Thanks!
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hobold
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« Reply #1 on: March 01, 2010, 11:42:36 AM »

The Mandelbrot set is indeed connected. This is a consequence of all the level sets being topologically equivalent to concentric circles. A sketch of the proof can be found in "The Science of Fractal Images" on page 189, for example.
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aluminumstudios
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« Reply #2 on: March 01, 2010, 04:33:14 PM »

Thanks for the reply.  Maybe I'll have to check out that book if I can find it on amazon.co.jp (I'm currently in Japan but Amazon's Japanese site has a lot of English books.)

I had a thought and want to make sure I am getting my terminology right.  When I say "connected", I mean for example that I could travel from inside the biggest mini-mandelbrot (at approx -1.75, 0) to the mini-mandelbrot at approx (-.007, 1.003) without ever leaving the "inside" of the set?


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hobold
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« Reply #3 on: March 01, 2010, 11:45:19 PM »

The Mandelbrot set is connected in the sense that from every minibrot, there is at least one thin line running all the way to the main body. That line might have points where it is so thin that it consists of only border, with no real inside (i.e. points around which you cannot center a disk with radius >0 that is completely contained in the set, no matter how small the radius).

An example of such a point would be where the largest disk just touches the main cardioid. If you move up or down from there, no matter how small the distance, you must leave the set.
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Timeroot
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« Reply #4 on: March 02, 2010, 05:06:22 AM »

...and all those "infinitely thin" points you'd have to travel through (which, by the way, occur in infinite number between the Mandelbrot and any minibrot, or between any two minibrots) are all either parabolic fixed points, where the period is increasing (these connect to "areas") or Misiurwecz (?) points, which map into unstable cycles; in other words, everything nearby is outside, and it's connected by (generally) spiral-shaped arms.
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