Weak Limits of the Measures of Maximal Entropy for Orthogonal Polynomials
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Weak Limits of the Measures of Maximal Entropy for Orthogonal Polynomials Carsten Lunde Petersen1
· Eva Uhre1
Received: 13 August 2019 / Accepted: 29 December 2019 / © Springer Nature B.V. 2020
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). Keywords Orthogonal polynomials · Julia set · Green’s function - Equilibriums measure Mathematics Subject Classification (2010) Primary 42C05; Secondary 37F10 · 31A15
1 Introduction and General Results In the classical study [5] by Stahl and Totik of general orthogonal polynomials they relate the potential and measure theoretic properties of the asymptotic zero distribution for the sequence of orthonormal polynomials defined by a Borel probability measure μ on C of infinite, but compact support S(μ), to the potential and measure theoretic properties of μ and its support. In the paper [4] Christiansen, Henriksen, Pedersen and one of the authors of this paper initiated a study of the relation between the potential theoretic properties of μ and the asymptotic (as n tend to ∞) potential and measure theoretic properties of the Julia sets and filled Julia sets of the orthonormal polynomials Pn . In this paper we extend this with a study of the weak convergence properties of the measures of maximal entropy for the orthonormal polynomials. For μ a Borel probability measure on C with infinite and compact support S(μ) we denote by {Pn (z)} := {Pn (μ; z)} the unique sequence Pn (z) = γn zn + lower order terms.
Carsten Lunde Petersen
[email protected] 1
IMFUFA at Department of Science and Environment, Roskilde University, 4000 Roskilde, Denmark
(1)
C.L. Petersen, E. Uhre
of orthonormal polynomials wrt. μ. Then Pn is also characterized as the unique normalized polynomial of the form Eq. 1 which is orthogonal to all lower degree polynomials or equivalently for which the monic polynomial pn (z) = Pn (z)/γn is the unique monic degree n polynomial of minimal norm in L2 (μ). For S ⊂ C a compact non-polar subset such as S(μ) above we denote by = (S) the unbounded connected component of CS, by K = K(S) := C the filled-in or just filled S (denoted by the polynomial convex hull in e.g. [5]), by J = J (S) = ∂K = ∂ ⊂ S the outer boundary of S, by g the Green’s function with pole at infinity for and finally by ωS the equilibrium measure for S (with Supp(ωS ) = J ). For a measure μ ∈ B we shall write K(μ) for K(S(μ)), J (μ) for J (S(μ)) and (μ) for (S(μ)) or simply K, J and , when the measure μ is understood from the context. Definitio
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