Minkowski dimension of the boundaries of the lakes of Wada

Zhangchi Chen

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arXiv · 2107.05626

Minkowski dimension of the boundaries of the lakes of Wada

Zhangchi Chen

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Abstract

The lakes of Wada are three disjoint simply connected domains in $S^2$ with the counterintuitive property that they all have the same boundary. The common boundary is a indecomposable continuum. In this article we calculated the Minkowski dimension of such boundaries. The lakes constructed in the standard Cantor way has $\ln(6)/\ln(3)\approx 1.6309$-dimensional boundary, while in general, for any number in $[1,2]$ we can construct lakes with such dimensional boundaries.

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Zhangchi Chen. 2021-07-12. Minkowski dimension of the boundaries of the lakes of Wada. https://arxiv.org/abs/2107.05626

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