THE SHARPNESS OF SOME CLUSTER SET RESULTS

We show that a recent cluster set theorem of Rung is sharp in a certain sense. This is accomplished through the construction of an interpolating sequence whose limit set Is c]osed, totally disconnected and porous. The results also generalize some of Dolzenko’s cluster set theorems.

INTERPOLATING SEQUENCES. We begin by considering a closed totally disconnected set P on the boundary 3a of the unit disc a in the complex plane. Thus =3a-P is the union of countably many disjoint arcs. Our first objective is to construct interpolating sequences on certain curves in the unit dlsc a whose llmlt points are all the points of P. In a speclal case Dolzenko [1] used this construction apparently not realizing that he was dealing wlth Interpolating sequences. We wish to define an approach to a point z a inside a reasonably nice subdomain of a. Let h(t) be a real-valued function defined for -1 < t < 1. We require that (1) h be continuous.    3) The first curve in (1.3) Is called the right h-curve at and the second curve in (1.3} is called the left h-curve at {See Fig. 1). Clearly these curves are rotations of the corresponding curves at 1.
We construct an Interpolating sequence which has P as its limit set.
Recall that a sequence {Zn} is an interpolating sequence if, for each bounded sequence of complex numbers {Wn }, there exists a function f in such that f{Zn}= n for every n. We shall use the characterization of Garnett  x(a,b} The right inequality iplies that if then 10) x(a,b) > 4--Thus we show that (1.9) is valid. Let zmn be an element of the sequence on the h-curve ending at n and let z k be an eleaent of the sequence on the h-curve ending at rj as shown in Fig. 1 1-1 al _< 5d.
(1.13) aesnD The points of S that lie in D belong to curves that end at the boundary of D except for at ost to curves which ight end outside D (See Fig. 1 1). Let z dent, re the first term of each sequence lying in D then where we used (1.5). Thus Sl (]-Ial) < 5d, a( D which implies that S is an interpolating sequence.

2.
POROSITY AND RIGHT h-ANGLES. In this section we add another restriction on the set P C BA. We assume that P is porous. The notion of porosity was introduced in 1967 by Dolzenko [1] and later used by Run [4] and Yoshida [5] to generalize some of the cluster theory results. We note that in 1976 Zajicek [6] generalized the definition of porosity and proved a variety of interesting properties of porous sets.
Let P C BA. For each e ioe a, let (e,e,P) be the length of the largest subarc of the arc (e i{o-e), e i(o+e)) which does not meet P. (rei#:l-ha(-O) < r < 1-hb(#-e), 0 < -e < a). (2. 2) The boundary curve of RA(0,a,b,h) defined by the left inequality will be called the lower boundary curve of the right h-angle domain and the other boundary curve is called the upper boundary curve (See Fig. 1) an___d 60 such that the set {z.lB(z)] < } is contained in the unioa of disjoint pseudohype[bolic discs N{an,50) wit_____h 1-center a n and I-radius 0" Theorem 1. Le___t P be a closed totally disconnected porous subset of aa and let h be a convex approach function. Then there exists a Blaschke function B{z) with the following properties" We now prove (li). If r=e i E a-P then (li) is obvious. If E P then we first show that (ii) holds for the case when r=e i is an isolate(| point of P. In this circumstance r is the initla] endpoint of an arc (r,r*) contained in a-P. Using   When the intervals approach 1 from above the left angle at is replaced by the corresponding right angle at and the proof proceeds along the same lines as before. This completes the proof of the theorem.