Weak Limits of the Measures of Maximal Entropy for Orthogonal Polynomials

Abstract

In this paper we study the sequence of orthonormal polynomials {P_n(μ;z)} defined by a Borel probability measure μ with non-polar compact support $S(\mu )\subset \mathcal {C}$. For each n ≥ 2 let ω_n denote the unique measure of maximal entropy for P_n(μ;z). We prove that the sequence {ω_n}_n is pre-compact for the weak-* topology and that for any weak-* limit ν of a convergent sub-sequence $\{\omega _{n_k}\}$, 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 {ω_n}_n converges weak-* to the equilibrium measure ω on S(μ).