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 e-prints
- Pub Date:
- July 2021
- DOI:
- 10.48550/arXiv.2107.05626
- arXiv:
- arXiv:2107.05626
- Bibcode:
- 2021arXiv210705626C
- Keywords:
-
- Mathematics - General Topology;
- Mathematics - Metric Geometry;
- 28A80;
- 28A78
- E-Print:
- 13 pages, 16 figures, Keywords: Hausdorff dimension, Minkowski dimension, Lakes of Wada