Weak Limits of the Measures of Maximal Entropy for Orthogonal Polynomials

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 ( μ ) ⊂ 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 { ω 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 ).