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

Let $A$ be a compact set in $\text{Rp}$ 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 $$\text{Is}(\mu ):=\iint \text{Rk}xysd\mu (y)d\mu (x)$$ 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.