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

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 $$ \Is(μ):=\iint\Rk{x}{y}{s}dμ(y)dμ(x)$$ 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.