Rational compacts and exposed quadratic irrationalities

Abstract

If {e(i)}(i=1)(g+1) are non-intersecting closed arcs on the unit circle T then their union E is called rational if all harmonic measures nu(E) (e(j)) at infinity are rational. It is known that the essential support supp(ess) (sigma) of a periodic measure sigma (i.e. the Verblunsky parameters of sigma are periodic) is rational and any rational E is a rotation of supp(ess) (sigma) for a periodic sigma. Elementary proofs of these facts are given. The Schur function f of a periodic sigma satisfies zA* f(2) + (B - zB*)f - A = 0, where the pair (A, B) of polynomials in z is called a Wall pair for sigma. Then supp(ess) (sigma) = {t is an element of T : vertical bar b+(t)vertical bar(2) <= 4 omega}, b(+) = B + zB*, omega = C(E)(2deg(b+)), C(E) being the logarithmic capacity of E. For any monic b with roots on T, b* = b, and omega satisfying 0 < 4 omega <= m(b)(2), where m(b) is the smallest local maximum of vertical bar b vertical bar on T, there is a Wall pair (A, B) such that b = B + zB* and supp(ess) (sigma) = {t is an element of T : vertical bar b(t)vertical bar(2) <= 4 omega} for any periodic sigma corresponding to (A, B). The solutions to the equation b = B + zB* in B related to Wall pairs are described. As a consequence we obtain the inverse Bernstein inequality for a separable polynomial b with roots on T: inf(T) vertical bar b'vertical bar >= 0.5 . m(b) . deg(b). The inequality is precise. A complete description of essential supports of periodic measures is also given in terms of the phases of Akhiezer's multi-valued analytic function as well as separable monic polynomials related to it with roots on T. (C) Elsevier Inc. All rights reserved.