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 ν E ( e j ) at ∞ are rational. It is known that the essential support supp e s s ( σ ) of a periodic measure σ (i.e. the Verblunsky parameters of σ are periodic) is rational and any rational E is a rotation of supp e s s ( σ ) for a periodic σ . Elementary proofs of these facts are given. The Schur function f of a periodic σ satisfies z A ∗ f 2 + ( B − z B ∗ ) f − A = 0 , where the pair ( A , B ) of polynomials in z is called a Wall pair for σ . Then supp e s s ( σ ) = { t ∈ T : | b + ( t ) | 2 ⩽ 4 ω } , b + = B + z B ∗ , ω = C ( E ) 2 deg ( b + ) , C ( E ) being the logarithmic capacity of E . For any monic b with roots on T , b ∗ = b , and ω satisfying 0 < 4 ω ⩽ m b 2 , where m b is the smallest local maximum of | b | on T , there is a Wall pair ( A , B ) such that b = B + z B ∗ and supp e s s ( σ ) = { t ∈ T : | b ( t ) | 2 ⩽ 4 ω } for any periodic σ corresponding to ( A , B ) . The solutions to the equation b = B + z B ∗ 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 | b ′ | ⩾ 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 .