Minkowski dimension of the boundaries of the lakes of Wada

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.

arXiv comment13 pages, 16 figures, Keywords: Hausdorff dimension, Minkowski dimension, Lakes of Wada