Riesz s-Equilibrium Measures on d-Dimensional Fractal Sets as s Approaches d

TL;DR

This paper proves that for self-similar fractals, Riesz s-equilibrium measures converge to the Hausdorff measure as s approaches the fractal's dimension from below.

Contribution

It establishes the convergence of Riesz s-equilibrium measures to Hausdorff measure on self-similar fractals as s approaches the dimension d.

Findings

Riesz s-equilibrium measures converge weak-star to Hausdorff measure as s approaches d.

The convergence holds for strictly self-similar d-fractals.

The result links potential theory with fractal geometry.

Abstract

Let $A$ be a compact set in $\Rp$ of Hausdorff dimension $d$. For $s\in(0,d)$, the Riesz $s$-equilibrium measure $\mu^{s,A}$ is the unique Borel probability measure with support in $A$ that minimizes $$ \Is(\mu):=\iint\Rk{x}{y}{s}d\mu(y)d\mu(x)$$ over all such probability measures. In this paper we show that if $A$ is a strictly self-similar $d$-fractal, then $\mu^{s,A}$ converges in the weak-star topology to normalized $d$-dimensional Hausdorff measure restricted to $A$ as $s$ approaches $d$ from below.