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.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.05626
 Bibcode:
 2021arXiv210705626C
 Keywords:

 Mathematics  General Topology;
 Mathematics  Metric Geometry;
 28A80;
 28A78
 EPrint:
 13 pages, 16 figures, Keywords: Hausdorff dimension, Minkowski dimension, Lakes of Wada