6
$\begingroup$

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

  • 0
    @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).2012-03-19

1 Answers 1

4

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)