Lakes of Wada

In mathematics, the lakes of Wada (和田の湖, Wada no mizuumi) are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary.

For example, the first five days might be (see the image on the right): 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.

Usually, the Wada property can be seen in the basin of attraction of dissipative dynamical systems.

But the exit basins of Hamiltonian systems can also show the Wada property.

M. A. F. Sanjuán et al.[1] has shown that in the Hénon–Heiles system the exit basins have this Wada property.

First five stages of the lakes of Wada
Animation of digging lakes up to day 5
Newton fractal forming Wada basins of attraction for z 3 − 1 = 0; all three disconnected open basins have the same boundary