Rational compacts and exposed quadratic irrationalities

If {ei}i=1g+1 are non-intersecting closed arcs on the unit circle TT then their union EE is called rational if all harmonic measures νE(ej)νE(ej) at ∞∞ are rational. It is known that the essential support suppess(σ) of a periodic measure σσ (i.e. the Verblunsky parameters of σσ are periodic) is rational and any rational EE is a rotation of suppess(σ) for a periodic σσ. Elementary proofs of these facts are given. The Schur function ff of a periodic σσ satisfies zA∗f2+(B−zB∗)f−A=0zA∗f2+(B−zB∗)f−A=0, where the pair (A,B)(A,B) of polynomials in zz is called a Wall pair for σσ. Then suppess(σ)={t∈T:|b+(t)|2⩽4ω}, b+=B+zB∗b+=B+zB∗, ω=C(E)2deg(b+)ω=C(E)2deg(b+), C(E)C(E) being the logarithmic capacity of EE. For any monic bb with roots on TT, b∗=bb∗=b, and ωω satisfying 0<4ω⩽mb2, where mbmb is the smallest local maximum of |b||b| on TT, there is a Wall pair (A,B)(A,B) such that b=B+zB∗b=B+zB∗ and suppess(σ)={t∈T:|b(t)|2⩽4ω} for any periodic σσ corresponding to (A,B)(A,B). The solutions to the equation b=B+zB∗b=B+zB∗ in BB related to Wall pairs are described. As a consequence we obtain the inverse Bernstein inequality for a separable polynomial bb with roots on TT: infT|b′|⩾0.5⋅mb⋅deg(b)infT|b′|⩾0.5⋅mb⋅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 TT.