Lakes of Wada
More than two sets with the same boundary are said to have the Wada property; examples include Wada basins in dynamical systems.
Construction of the lakes of Wada
The Lakes of Wada are formed by starting with an open unit square of dry land (homeomorphic to the plane), and then digging 3 lakes according to the following rule:
- On day n = 1, 2, 3,... extend lake n mod 3 (=0, 1, 2) so that it passes within a distance 1/n of all remaining dry land. This should be done so that the remaining dry land has connected interior, and each lake is open.
After an infinite number of days, the three lakes are still disjoint connected open sets, and the remaining dry land is the boundary of each of the 3 lakes.
For example, the first five days might be (see the image on the right):
- Dig a blue lake of width 1/3 passing within √2/3 of all dry land.
- Dig a red lake of width 1/32 passing within √2/32 of all dry land.
- Dig a green lake of width 1/33 passing within √2/33 of all dry land.
- Extend the blue lake by a channel of width 1/34 passing within √2/34 of all dry land. (Note the small channel connecting the thin blue lake to the thick one, near the middle of the image.)
- Extend the red lake by a channel of width 1/35 passing within √2/35 of all dry land. (Note the tiny channel connecting the thin red lake to the thick one, near the top left of the image.)
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.
Wada basins are certain special basins of attraction studied in the mathematics of non-linear systems. A basin having the property that every neighborhood of every point on the boundary of that basin intersects at least three basins is called a Wada basin, or said to have the Wada property. Unlike the Lakes of Wada, Wada basins are often disconnected.
An example of Wada basins is given by the Newton–Raphson method applied to a cubic polynomial with distinct roots, such as z3 − 1; see the picture.
A physical system that demonstrates Wada basins is the pattern of reflections between three spheres in contact—see chaotic scattering.
- Breban, Romulus; Nusse, H E. (2005), "On the creation of Wada basins in interval maps through fixed point tangent bifurcation", Physica D 207 (1–2): 52–63, doi:10.1016/j.physd.2005.05.012
- Coudene, Yves (2006), "Pictures of hyperbolic dynamical systems", Notices of the American Mathematical Society 53 (1): 8–13, ISSN 0002-9920, MR2189945
- Gelbaum, Bernard R.; Olmsted, John M. H. (2003), Counterexamples in analysis, Mineola, N.Y.: Dover Publications, ISBN 0-486-42875-3 example 10.13
- Hocking, J. G.; Young, G. S. (1988), Topology, New York: Dover Publications, p. 144, ISBN 0-486-65676-4
- Kennedy, J; Yorke, J.A. (1991), "Basins of Wada", Physica D 51: 213–225, doi:10.1016/0167-2789(91)90234-Z
- Sweet, D.; Ott, E.; Yorke, J. A. (1999), "Complex topology in Chaotic scattering: A Laboratory Observation", Nature 399 (6734): 315, doi:10.1038/20573
- Yoneyama, Kunizô (1917), "Theory of Continuous Set of Points", Tôhoku Mathematical Journal 12: 43–158