Computing the equilibrium measure of a system of intervals converging to a Cantor set
Provides a practical computational tool for potential theory on fractal sets, which is a niche domain-specific problem.
The paper develops a numerical method to compute the equilibrium measure on Cantor-like attractors of iterated function systems, using limits of measures on interval unions. The method handles numerical instabilities and demonstrates convergence via electrostatic potential calculations.
We describe a numerical technique to compute the equilibrium measure, in logarithmic potential theory, living on the attractor of Iterated Function Systems composed of one-dimensional affine maps. This measure is obtained as the limit of a sequence of equilibrium measures on finite unions of intervals. Although these latter are known analytically, their computation requires the evaluation of a number of integrals and the solution of a non-linear set of equations. We unveil the potential numerical dangers hiding in these problems and we propose detailed solutions to all of them. Convergence of the procedure is illustrated in specific examples and is gauged by computing the electrostatic potential.