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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

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    $\begingroup$ 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... $\endgroup$ – Jason DeVito Mar 19 '12 at 15:55
  • $\begingroup$ 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 ..." $\endgroup$ – Patrick Mar 19 '12 at 15:57
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    $\begingroup$ 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. $\endgroup$ – lhf Mar 19 '12 at 16:49
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    $\begingroup$ Jason, separability precludes finding any uncountable collection of pairwise disjoint nonempty open sets, irrespective of any conditions you put on their boundaries. $\endgroup$ – user83827 Mar 19 '12 at 17:23
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    $\begingroup$ @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) $\endgroup$ – lhf Mar 19 '12 at 17:31
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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)
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