T HE UN IV ERSIT Y OF MI CHI GAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of' Mathematics Technical Note ASYMMETRIC PRIME ENDS E. F. Cq~.1ingwood G.?faian UMRI Project 29135 under contract with: MATHEMATICAL SCIENCES DIRECTORATE AIR' FORCE OFFICE OF SCIENTIFIC RESEARCH,CONTRACT NO..AF 4-i9(638) —633'WASHINGTON 25, D. C. administered by THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR

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SUMMARY Each simply connected domain in the plane has at most countably many prime ends whose right and left wings do not coincide. On the other hand, to each countable set E on the unit circle C there corresponds a function which is holomorphic and univalent in the unit disk D and which has the property that it carries each point of E and no point of C E onto a prime end with unequal wings ii

1. CLUSTER SETS Let f denote a function which maps the unit disk D: JzJ < 1 onto the Riemann sphere R, and let eig be a fixed point on the unit circle C: Izl = 1. A point w on R belongs to the cluster set C(f, ei) of f at ei0 provided there exists a sequence [zn) = (rneign) such that rnt l, Gn + @, f(zn) + w. It belongs to the radial cluster set Cp(f, ei@) provided the sequence [Zn) can be chosen with On = 9 for all n. It belongs to the right cluster set CR(f, eiG) (or to the left cluster set CL(f, eig)) provided the sequence (Zn} can be chosen with n1/1A (or with Gn 0). If f is continuous in D, then the set of points eig where Cp(f, eig) does not contain C(f, eig) is a set of first category [1]; but even if f is required to be holomorphic and univalent in D, the point set in question may have measure 2x, as can be seen, for example, from the construction described in [4, Section 2]. In other words, the set where the radial cluster set is a proper subset of the complete cluster set is thin topologically but may be large geometrically. The following theorem implies that the point set where the right and left cluster sets do not coincide is subject to much more severe restrictions, even if no conditions whatever are imposed on the function f. THEOREM 1. If f maps the unit disk into the Riemann sphere, then CR(f, eiQ) = C(f, eig) = CL(f, eiG) for all except at most countably many points eiG.

Since the right (left) cluster set of a function contains the set which has been called the right (left) boundary cluster set, Theorem 1 is a corollary of Collingwood's recent theorem [2] to the effect that the right and left boundary cluster sets at ei0 coincide with C(f, eig), except possibly at countably many points eig. However, for the sake of completeness and simplicity, we give here a proof which is independent of the concept of boundary cluster sets. We cover the Riemann sphere R with a succession of nets NK (k = 1,2,...), each with finitely many (closed) triangular meshes mkn(n = 1, 2,..., nk) of diameter less than l/k. By means of any convenient ordering, we arrange all the meshes mkn into a sequence (mj). For each index j, we consider the set Sj of those points eig for which the mesh mj contains a point of C(f, ei0) but does not meet the set CR(f, eig). Clearly, the set SR of points eig for which C(f, eiG)NCR(f, eig) is not empty is the union of the sets Sj. If eig ~ Sj, then eiG is the endpoint of an arc I(@) = (ei, eig)(0 < such that C(f, eiV) does not meet the mesh mj if ei4 lies on I(G); for otherwise mi would meet the set CR(f, ei), contrary to the hypothesis that ei E: Sj. Since the arc I(G) cannot contain points of Sj, the set Sj, is at most countable, and therefore SR is at most countable. Likewise, the corresponding set SL is at most countable, and the theorem is proved. 2. ASYMMETRIC PRIME ENDS Let the function f be holomorphic and univalent in the unit disk D, and let B denote the image of D under f. If P is the prime end of B that corresponds to the point eiG, the two cluster sets CR(f, eiG) and CL(f, ei0) coincide with

what Ursell and Young [6, p. 14] have called the right and left wings of PO We shall say that the prime end P is asymmetric if its two wings are not identical. The following theorem is an immediate consequence of Theorem 1o THEOREM 2. Each simply connected plane domain has at most countably many asymmetric prime ends. In order to formulate our next theorem, we partition the set of asymmetric prime ends of a fixed simply connected domain into three classes UR, UL, and URLI as follows: a prime end belongs to UR if its left wing is a proper subset of its right wing, to UL if its right wing is a proper subset of its left wing, and to URL if neither of its wings is a subset of the other. THEOREM 3. Let ER, EL, and ERL be three disjoint sets on the unit circle C, each at most countable. Then there exists a function f which maps the unit disk conformally onto a domain B in such a way that each point in ER, EL, or ERL corresponds to a prime end in UR, UL, or URL, respectively, while all other points of C correspond to symmetric prime ends of B. To prove Theorem 3, we order the points of ERU ELU ERL into a sequence (zj) (j = 1, 2,.. )o With each point Zj we associate a sequence of points zjp lying on C and converging to zj. We construct a function of the form f(z) = z + Ajp(l - (1 - z/pJpzjp) P ) j,P where the Ajp are appropriate complex constants, kzjp O and Pjp4l as + O, and where the symbol in braces represents that branch of the corresponding function which takes the value O at z = 0. 5

The constants that determine the function f can be chosen in such a way that f is holomorphic and univalent in D, such that the radial limit f(eiG) of f exists, for each value 0, and such that the boundary of the image of D, in the neighborhood of a point f(zj), appears roughly as in Figure 1, 2, or 3, according as zj belongs to ER, EL, or ERL. We say "roughly" because each of the tooth-like extensions of the domain B may carry further extensions (indeed, the deformations may be everywhere dense on the boundary; difficulties that might arise from unwanted condensations of singularities can be avoided by subjecting the constants Ajp to the restriction IAjpl < l/j:, in other words, by requiring that the teeth associated with the point zj have length at most l/j'. The details of the proof are similar to the details in [3, Sections 3 and 4], and we omit them. Theorem 3 makes no mention of Caratheodory's four kinds of prime ends. Our construction leads to a domain whose asymmetric prime ends are of the second kind and whose symmetric prime ends are of the first kind. No matter what construction is used, the asymmetric prime ends must be of the second or fourth kind, since the prime ends of the first and third kinds have no subs~Ldary points The following theorem shows that no other topological restriction on the distribution of the asymmetric prime ends exists. THEOREM 4. Let C = ElUE2UE3U E4 be a decomposition of the unit circle such that some homeomorphic mapping of the unit disk onto a simply connected domain B induces a homeomorphism ~ of the sets E1, E2, E3, E4 onto the sets of prime ends of the first, second, third, and fourth kind, respectively, of B. Let SR, SL and SRL be disjoint countable subsets of E2U E4. Then the domain

(D H-~~~~~~ ei /~~ ~H \J1@ -~~~~~~~~~~~~~~ R) /~

B can be chosen in such a way that SR, SL, and SRL correspond to the sets UR, UL, and URL, respectively, under the homeomorphism ~. A detailed proof of Theorem 4 would be tedious, and we give only a brief sketch to show how the construction in Sections 3 to 8 of [5 ] can be modified so as to yield the desired result. In [5], the set E234 of points corresponding to prime ends of the second, third, or fourth kind is represented as the union of a sequence of disjoint sets: E234.i (i = 1, 2,...). The set E234.1 is open, and each of the sets E234.i (i = 2, 3,...) is closed and nowhere dense. The unit disk is subjected to certain deformations determined by the set E234.2; the process is repeated (on a successively smaller scale) with reference to E234.i (i = 3, 4,...). Thereafter, a special program of further deformations is launched with reference to E234.1. For our present purpose, we denote by SR. i SL. i' and SRL.i the intersections of SR, SL, and SRL with E234.i. First we order the set SR.2USL.2 USRL.2 into a finite or infinite sequence (Z2j ). In the neighborhood of z21, we deform the boundary of the unit disk in the manner indicated by Figures 4, 5, or 6, according as Z21 belongs to SR, SL, or SRL, and we map the set CN\z21 onto the curved portion of the deformed boundary. We proceed similarly (but on smaller scales) with regard to z22, z23,..-. The unit disk is then mapped onto the star-domain which has been obtained, in such a way that the radii.of D are deformed as is indicated in Figure 7. Clearly, the transformation can be constructed in such a way that it is continuous throughout the disk 3zj < 3/4 and on every radius of D that does not terminate at one of the points Z2 j. The 6

Z21 Z2 Figure 4 Figure 5 z21 Figure 6 Figure 7

transformed unit disk is then further deformed according to the program in Sections 5 and 6 of [5]. The analogous deformations which have to be carried out with reference to the sets E234.i (i = 3, 4,...) are obvious (see Section 7 of [5]). The deformations of the disk D that are used in [5] for the sake of the set E234 are somewhat more complicated, because E234.1 consists of interior points. However, the intersection of 2U E4 with E234.1 is partitioned into sets that are nowhere dense, and each of these is treated separately. Therefore we can proceed as above, provided we exercise one precaution: In the third and fourth paragraphs of Section 8 of [5], a certain denumerable set and a certain near-perfect set are extracted from E2 E234.1 and used for special purposes. These special purposes would interfere with our technique of creating asymmetric prime ends at the points of the extracted sets, and therefore the extracted sets must be selected in such a way that they contain no points of SR, SL, or SRL. Since E2 is locally uncountable in E234.1, the latter requirement presents no special difficulty. This sketch must suffice, because any exposition which covered all details would have to include the laborious description in [5]; and it is easier for the reader to acquaint himself first with the description in [5], and to supply thereafter the modifications which we have now sketched, than it would be to struggle through a detailed description in which the various operations are presented simultaneously. 8

REFERENCES 1. E. F. Collingwood, Sur le comportement a la frontiere, d'une fonction meromorphe dans le cercle unite, C. R. Acad. Sci. Paris 240 (1955), 15021504. 2. E. F. Collingwood, Cluster sets of arbitrary functions (to appear). 3. F. Herzog and G. Piranian, Sets of convergence of Taylor series, II, Duke Math. J. 20 (1953), 41-54. 4. A. J. Lohwater and G. Piranian, The boundary behavior of functions analytic in a disk, Ann. Acad. Sci. Fenn. Ser. A. I. no. 239 (1957), 1-17. 5. G. Piranian, The distribution of prime ends, Michigan Math. J. 7 (1960), 83-95. 6. H. D. Ursell and L. C. Young, Remarks on the theory of prime ends, Mem. Amer. Math. Soc. no. 3 (1951), 1-29. 9

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