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I've wondered about the following question, whose answer is perhaps well known (in this case I apologize in advance).

The Lakes of Wada are a famous example of three disjoint connected open sets of the plane with the counterintuitive property that they all have the same boundary (!)

My question is the following :

Can we find four disjoint connected open sets of the plane that have the same boundary?

More generally :

For each $n \geq 3$, does there exist $n$ disjoint connected open sets of the plane that have the same boundary? If not, then what is the smallest $n$ such that the answer is no?

Thank you, Malik

Malik Younsi
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    The article you link to says "A variation of this construction can produce a countable infinite number of connected lakes with the same boundary: instead of extending the lakes in the order 1, 2, 0, 1, 2, 0, 1, 2, 0, ...., extend them in the order 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, ...and so on.", so the answer seems to be "you can do it for all $n$ including $n =\aleph_0$". I'd guess (but have no idea really) that the corresponding question for other cardinalities is independent of ZFC, assuming NOT CH. Though, since $\mathbb{R}^n$ is first countable, there may be an answer... – Jason DeVito - on hiatus Mar 19 '12 at 15:55
  • Found this in the link. It seems to answer your question. "A variation of this construction can produce a countable infinite number of connected lakes with the same boundary: instead of extending the lakes in the order 1, 2, 0, 1, 2, 0, 1, 2, 0, ...., extend them in the order ..." – Patrick Mar 19 '12 at 15:57
  • See, also, here, where your question is answered in the affirmative. – David Mitra Mar 19 '12 at 16:05
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    Given that the basins of attraction of Newton's method for $z^3=1$ are an example for $n=3$, I'd say that the basins of attraction of Newton's method for $z^n=1$ should work just as fine. – lhf Mar 19 '12 at 16:49
  • @lhf Your comment is very interesting! It should be clear that for $n=3$ the basins of attraction of Newton´s method for $z^n=1$ are an example? Otherwise may I have a reference or an idea for the proof? Actually I cannot see it! – Giovanni De Gaetano Mar 19 '12 at 17:06
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    Jason, separability precludes finding any uncountable collection of pairwise disjoint nonempty open sets, irrespective of any conditions you put on their boundaries. – user83827 Mar 19 '12 at 17:23
  • @Student73, I don't know a reference except Wikipedia. Perhaps there's something in Newton's method and dynamical systems, edited by Heinz-Otto Peitgen, Reprint from Acta applicandae mathematicae, vol. 13:1-2. – lhf Mar 19 '12 at 17:27
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    @Student73, ah, this may contain a proof: Frame, Michael; Neger, Nial. Newton's method and the Wada property: a graphical approach. College Math. J. 38 (2007), no. 3, 192–204. MR2310015 (2008e:37082) – lhf Mar 19 '12 at 17:31
  • @lhf Thank you! I´m going to read it! – Giovanni De Gaetano Mar 19 '12 at 17:41
  • Oh... I missed that sentence in the wikipedia link (for some reason, it didn't display correctly on my phone...) Anyway, thank you all for your comments! @lhf : I didn't know about the basins of attraction approach to the lakes of Wada, that's very interesting. If you're willing to post that link as an answer, I will gladly accept it! – Malik Younsi Mar 19 '12 at 18:13
  • @ccc: You're right - thanks for catching that. (Also, you need to put the @ in front of my name in order for me to get pinged about it. I just randomly came back to this question). – Jason DeVito - on hiatus Mar 19 '12 at 19:19

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Given that the basins of attraction of Newton's method for $z^3=1$ are Wada sets for $n=3$, I'd say that the basins of attraction of Newton's method for $z^n=1$ should work just as fine. I don't know a reference to a proof but try these:

  • Newton's method and dynamical systems, edited by Heinz-Otto Peitgen, Reprint from Acta applicandae mathematicae, vol. 13:1-2.
  • Frame, Michael; Neger, Nial. Newton's method and the Wada property: a graphical approach. College Math. J. 38 (2007), no. 3, 192–204. MR2310015 (2008e:37082)
lhf
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