Computing the equilibrium measure of a system of intervals converging to a Cantor set

TL;DR

This paper introduces a numerical method to compute the equilibrium measure on fractal-like sets generated by iterated function systems, addressing computational challenges and demonstrating convergence through examples.

Contribution

It presents a detailed numerical technique for calculating equilibrium measures on Cantor-like sets, including solutions to potential computational issues.

Findings

The method accurately computes equilibrium measures on complex fractal sets.

Convergence is demonstrated through specific examples and electrostatic potential calculations.

The approach addresses and overcomes numerical challenges in integral evaluation and nonlinear equations.

Abstract

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.