Weak limits of the measures of maximal entropy for Orthogonal polynomials

In this paper we study the sequence of orthonormal polynomials ${P_n(\mu; z)}$definedbyaprobabilitymeasure$\mu$withnon-polarcompactsupport$S(\mu)\subset\mathbb C$. 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 $S(\mu )$.