TITLE: Does there exist more than 3 connected open sets in the plane with the same boundary?
QUESTION [6 upvotes]: 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

REPLY [4 votes]: 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)