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

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