Weak limits of the measures of maximal entropy for Orthogonal polynomials

In this paper we study the sequence of orthonormal polynomials {Pn(μ;z)} defined by a Borel probability measure μ with non-polar compact support S(μ)⊂C. For each n ≥ 2 let ωn denote the unique measure of maximal entropy for Pn(μ;z). We prove that the sequence {ωn}n is pre-compact for the weak-* topology and that for any weak-* limit ν of a convergent sub-sequence {ωnk}, the support S(ν) is contained in the filled-in or polynomial-convex hull of the support S(μ) for μ. And for n-th root regular measures μ the full sequence {ωn}n converges weak-* to the equilibrium measure ω on S(μ).