Periodic Schur functions and slit discs

A simply connected domain GG is called a slit disc if G=DG=D minus a finite number of closed radial slits not reaching the origin. A slit disc is called rational (rationally placed) if the lengths of all its circular arcs between neighboring slits (the arguments of the slits) are rational multiples of 2π2π. The conformal mapping ϕϕ of DD onto GG, ϕ(0)=0ϕ(0)=0, ϕ′(0)>0ϕ′(0)>0, extends to a continuous function on TT mapping it onto ∂G∂G. A finite union EE of closed non-intersecting arcs ekek on TT is called rational if νE(ek)∈QνE(ek)∈Q for every kk, νE(ek)νE(ek) being the harmonic measures of ekek at ∞∞ for the domain C∖EC∖E. A compact EE is rational if and only if there is a rational slit disc GG such that E=ϕ−1(T)E=ϕ−1(T). A compact EE essentially supports a measure with periodic Verblunsky parameters if and only if E=ϕ−1(T)E=ϕ−1(T) for a rationally placed GG. For any tuple (α1,…,αg+1)(α1,…,αg+1) of positive numbers with ∑kαk=1∑kαk=1 there is a finite family {ek}k=1g+1 of closed non-intersecting arcs ekek on TT such that νE(ek)=αkνE(ek)=αk. For any set E=∪k=1g+1ek⊂T and any ϵ>0ϵ>0 there is a rationally placed compact E∗=∪k=1g+1ek∗ such that the Lebesgue measure |E△E∗||E△E∗| of the symmetric difference E△E∗E△E∗ is smaller than ϵϵ.