Minkowski dimension of the boundaries of the lakes of Wada

TL;DR

This paper calculates the Minkowski dimension of the common boundary of Wada lakes, showing it can be precisely determined for lakes constructed via standard Cantor methods and generalized for any dimension in [1,2].

Contribution

It provides the first explicit calculation of the Minkowski dimension of Wada lake boundaries and generalizes the construction to any dimension in [1,2].

Findings

Wada lakes have a Minkowski boundary dimension of approximately 1.6309 with standard Cantor construction.

The boundary dimension can be generalized to any value in [1,2].

The boundary is an indecomposable continuum.

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.