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Mandelbrot set is connected. That is to say within a mandelbrot set for any pair of points there is a path within the set, connecting these points.

What abouthe set of all other points? Is there a pair of points that cannot be connected without going through the Mandelbrot set?

sixtytrees
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1 Answers1

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Yes, it is connected. Adriand Douady and John H. Hubbard proved that the complement of the Mandelbrot set and $\{z\in\Bbb C\mid|z|>1\}$ are conformally equivalent. In particular, they are homeomorphic. So, since $\{z\in\Bbb C\mid|z|>1\}$ is connected, the complement of the Mandelbrot set is connected too.

José Carlos Santos
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    Isn't it a duplicate? The result of Douady/Hubbard is mentioned in the question that I mentioned above, and a sketch of the proof in the answer. – Martin R Jul 15 '20 at 08:37
  • @MartinR I don't agree that that makes it a duplicate. It is a different question. – José Carlos Santos Jul 15 '20 at 09:13
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    Is it? OP asks if the complement of the Mandelbrot set is connected. That is equivalent to the Mandelbrot set being simply connected, or equivalent to the complement having no bounded component. That is – as far as I can see – answered in the other Q&A. Or am I misunderstanding something? – Martin R Jul 15 '20 at 09:18
  • @MartinR I surely agree that the answer to the other question also answers this question. However, it requires the knowledge that asserting that a bounded subset of $\Bbb C$ is simply connected is equivalent to asserting its complement is connected, which is something which, although true, is far from obvious. – José Carlos Santos Jul 15 '20 at 09:28
  • Well, you don't need the “simply connected” part. OP asks if the complement is connected, and that means that it has no bounded component (and a simple answer using the maximum modulus principle is given in the other thread). – Anyway, It is not that important to me that I will further fight for it :) – Martin R Jul 15 '20 at 09:33
  • @MartinR I want to call your attention to this question, to which you have provided an answer (+1). It almost got closed as a PSQ. I disagree and I wrote against closing it at the CURED chatroom. – José Carlos Santos Jul 15 '20 at 09:37
  • Thanks (I had seen that in CURED). I must admit that it is at least borderline missing context. I have left a comment, suggesting that OP improves the question. Otherwise there is not much that can be done, I am afraid. – Martin R Jul 15 '20 at 09:41
  • @JoséCarlosSantos I've voted to reopen. – Gabriel Romon Jul 18 '20 at 06:43