Lakes of the Wada
The lakes of Wada ( English Lakes of Wada ) are a counterintuitive example in mathematics , more precisely in topology , of three disjoint connected open subsets of the plane with a common edge . The name of this example goes back to the Japanese mathematician Takeo Wada .
Construction of the lakes
You start with an open piece of land on the plain and dredge into these three lakes according to the following rules:
- On day n = 1, 2, 3… enlarge lake number n mod 3 so that its bank is nowhere further than 1 / n from the bank of another lake. The country should keep its interior connected to the path and every lake should remain open.
After an infinite number of days, the three lakes are still disjoint open sets and the rest of the land is the topological edge of all three lakes.
This is possible, for example, in the following way (see figure on the right):
Start with the open unit square {(x, y) | 0 <x <1, 0 <y <1} as the country (gray in the figure).
- Create a blue lake B with width 1/3 and length 2/3, around which a strip of land with width 1/3 remains: B = {(x, y) | x <2/3; 1/3 <y <2/3}. The blue lake is a maximum of √2 / 3 from any land point.
- Dredge a red lake 1/3 2 wide in the rest of the land , around which a strip of land 1/3 2 wide remains. The red lake is a maximum of √2 / 3 2 from any land point .
- Dredge a green lake 1/3 3 wide into the rest of the land , around which a strip of land 1/3 3 wide remains. The green lake is a maximum of √2 / 3 3 from any land point .
- Extend the blue lake with a channel 1/3 4 wide , around which a strip of land 1/3 4 wide remains. To make the lake coherent, a narrow connection is created between the original blue lake and the blue channel (visible in the middle of the picture). The extended blue lake is a maximum of √2 / 3 4 from any land point .
- Extend the red lake by a channel 1/3 5 wide , around which a strip of land 1/3 5 wide remains. A narrow channel connects the thin red channel to the large original red lake (visible near the upper left corner of the picture.) The extended red lake is a maximum of √2 / 3 5 from any land point .
This construction is continued analogously.
annotation
More than 3 lakes with the same limit are possible. By varying this construction, any finite number k of lakes with the same limit can be created. Instead of expanding the lake with the number i congruent n mod 3 on the nth day, the lake with the number i is expanded congruent n mod k.
You can even create a countable, infinite number of lakes with the same limit: Instead of the lakes in the order 0, 1, 2, 0, 1, 2, 0, 1, 2, 0…. you can expand them in the order 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 5 ... can be created or expanded.
literature
- Romulus Breban, HE Nusse: On the creation of Wada basins in interval maps through fixed point tangent bifurcation . In: Physica D: Nonlinear Phenomena . tape 207 , no. 1–2 , 2005, pp. 52-63 , doi : 10.1016 / j.physd.2005.05.012 .
- Yves Coudene: Pictures of hyperbolic dynamical systems . In: Notices of the American Mathematical Society . tape 53 , no. 1 , 2006, ISSN 0002-9920 , p. 8–13, ( digital version [PDF; 3.4 MB ]).
- Bernard R. Gelbaum, John MH Olmsted: Counterexamples in analysis . Dover Publications, Mineola NY 2003, ISBN 0-486-42875-3 (Example 10.13).
- John G. Hocking, Gail S. Young: Topology . Reprint of Reading, Massachusetts edition, 1961. Dover Publications, New York NY 1988, ISBN 0-486-65676-4 , pp. 144 .
- Judy Kennedy, James A. Yorke : Basins of Wada . In: Physica D: Nonlinear Phenomena . tape 51 , no. 1-3 , 1991, pp. 213-225 , doi : 10.1016 / 0167-2789 (91) 90234-Z .
- David Sweet, Edward Ott , James A. Yorke: Topology in chaotic scattering . In: Nature . tape 399 , no. 6734 , 1999, p. 315-316 , doi : 10.1038 / 20573 .
- Kunizô Yoneyama: Theory of Continuous Set of Points . In: The Tôhoku Mathematical Journal. 1st Series . tape 12 , 1917, ISSN 0040-8735 , p. 43–158 ( digitized version ).