# Weak limits of the measures of maximal entropy for Orthogonal polynomials

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## Abstract

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).
Translated title of the contribution Om svage grænser af målene med maksimal entropi for orthogonale polynomier English Potential Analysis 54 2 219-225 7 0926-2601 https://doi.org/10.1007/s11118-019-09824-5 Published - Feb 2021

### Bibliographical note

Important note from the Publisher regarding te attached version of the article: “This is a post-peer-review, pre-copyedit version of an article published in Potential Analysis. The final authenticated version is available online at: http://dx.doi.org/10.1007/s11118-019-09824-5”.

## Keywords

• Green’s function - Equilibriums measure
• Julia set
• Orthogonal polynomials
• ### Holomorpic Dynamics and Orthogonal Polynomials

Petersen, C. L., Petersen, H. L., Henriksen, C. & Christiansen, J. S.

01/11/201531/10/2018

Project: Research