Riesz s-equilibrium measures on d-dimensional fractal sets as s approaches d

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