Weak Limits of the Measures of Maximal Entropy for Orthogonal Polynomials

Abstract

In this paper we study the sequence of orthonormal polynomials $\{P_n(\mu; z)\}$ defined by a probability measure $\mu$ with non-polar compact support $S(\mu)\subset\mathbb C$. We show that the support of any weak* limit of the sequence of measures of maximal entropy $\omega_n$ for $P_n$ is contained in the polynomial-convex hull of $S(\mu)$. And for $n$-th root regular measures the $\omega_n$ converge weak* to the equilibrium measure on $S(\mu)$.