— THE a-SEGMENT PROPERTY FOR CQNHWGNSE WNCWGNS BEFMED N THE ELEWER EAL? PLANE The“; few- fho Dogma of N pk. D. MlCBiGAN SYATE UNIVERSITY Charis}: Leanaré Belna E 9 69 h‘ iumm .4 mt :Mt ‘§ LIBRAPV ““2 ) Michigan New: University This is to certifg that the thesis entitled The n-Segment Property for Continuous Functions Defined in the Upper Half Plane presented by Charles Leonard Belna has been accepted towards fulfillment of the requirements for Ph.D. (kgweinMathematics M— Major professor é ABSTRACT THE n-SEGMENT PROPERTY FOR CONTINUOUS FUNCTIONS DEFINED IN THE UPPER HALF PLANE By Charles Leonard Belna Let f be a function from the upper half plane H into the Riemann sphere W and let p be a point on the real line R. Then f is said to possess the n-segment property at p if there exist n n-segments Sl(p),...,Sn(p) at p such that O CSj(p) = ¢, where '=1 CSj(p) is the cluster set of f at p along JSj(p)u It is shown that the set of such points is countable if f is a homeomorphism of H onto a Jordan domain U° Various techniques which hinge on the Lebesgue Density Theorem are used to establish the following main results“ The set of points at which a continuous function has the n—segment property relative to either n or n-l fixed directions is an F0 set of first category and measure zero“ The set of points at which a continuous function has the nasegment property relative to n fixed points in H is an F6 set of first category and measure zero. Finally, the set of points at which a continuous function has the essential-n-segment property relative to n fixed directions is of measure zero. Concerning local behavior, the following are proved. A point at which a meromorphic function possesses the 3-segment property is A: Charles Leonard Belna in some sense "very close" to being an ambiguous point. There exists a holomorphic function having the strong 3-segment property at p = 0 which does not have the strong n-segment property at 0 for any other n 2 2. Also, if the principal cluster set is empty at a point p, then either p is an ambiguous point or the infimum of the diameters of all are cluster sets at p is positive, THE n-SEGMENT PROPERTY FOR CONTINUOUS FUNCTIONS DEFINED IN THE UPPER HALF PLANE By Charles Leonard Belna A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1969 ACKNOWLEDGMENT I wish to thank Professor Peter Lappan for suggesting the investigation of the are properties and for his patient guidance throughout the preparation of this thesis. ii I. II. III. IV. TABLE OF CONTENTS INTRODUCTION ............................................. 1 THE n-SEGMENT PROPERTY RELATIVE TO k DIRECTIONS .......... 9 THE n-SEGMENT PROPERTY RELATIVE TO n FIXED POINTS ........ 20 THE ESSENTIAL-n-SEGMENT PROPERTY RELATIVE TO n DIRECTIONS ............................................... 24 BEHAVIOR AT POINTS WHERE THE ARC PROPERTIES OCCUR ........ 30 BIBLIOGRAPHY“ ............................................. 37 iii I° INTRODUCTION The following notations will be used throughout this paper: With Z denoting the finite complex plane, we set C = {z E Z : lzl = l} , D = {z E Z : lzl < l} , H = {z E Z : Im(z) > O} , R = {z E Z : Im(z) = 0} , and R+ = {p E R : p > O} . Let f be a function from the upper half plane H into the Riemann sphere W and let p be a point on the real line R. The cluster Egg, C(f,p), E: f a; p is defined to be the set of all points w E W for which there exists a sequence {2k} of points in H with zk v p and f(zk) a w. If C is an arc in H such that C U {p} is a Jordan arc with one endpoint at p, then G is said to be an EEE pp p and the cluster ESE, C(f,p,o),.9£ f fig p 31225 o is defined to be the set of all points w E W for which there exists a sequence {2k} of points in o with zk n p and f(zk) a w. We say that f possesses the n-ggg propertx 33 p, for some integer n 2 2, if there exist n arcs 01,02,...,0n at p such that the intersection of all n of the sets C(f,p,01),C(f,p,02),...,C(f,p,0n) is empty. If, in addition, the intersection of any n71 of these n sets is non-empty we say that f possesses the strong n—arc property 55 p° Similarly, if the n 1 arcs can be chosen to be rectilinear segments, we say that f possesses the n—segment property or, correspondingly, the strong n-segment property 33 p. A point p E R at which f has the 2-arc property is called an ambiguous point 9f f. Analogous definitions are made for functions from the open unit disk D into W and for points p on the unit circle C. In 1955, Bagemihl [1] proved the following theorem which either sharpened or generalized various reSults of Blumberg [4], Schmeiser [21], and Jarnik [11]. BAGEMIHL'S AMBIGUOUS POINT THEOREM. IRE EEE pf ambiguous .l points 2f 32 arbitrary function from D into W is a3 most count- able. A few years ago, Mathews [16] noted that Bagemihl's theorem remains true when ”ambiguous points” is replaced by "extended ambiguous points". A point p E C is an extended ambiguous point of a function f if there exist two arcs o and U lying in 1 2 D - {p} except for one endpoint at p such that ‘Ec n Ec = <23 , where EC(f,p,oj) is the extended arc cluster set of f at p along cj which is defined analogous to the left and right boundary cluster sets. In a lecture presented at the Thirteenth Congress of Scandina- vian Mathematicians in 1957, Lohwater [13] proposed the investi- gation of the n-arc property for n > 2. Up to that time, only a few results pertaining to this more general property had been obtained: A result of Lindelgf [12, p. 28] in 1915 implied that for each n 2 2, a schlicht function possesses the n-arc property nowhere. In 1919, Gross [10, po 57, Section 20] gave a partial extension of Lindelgf's result to meromorphic functions which furnished the corollary that a meromorphic function from D into W does not possess the n-arc property for any n 2 2 at a point p E C where the cluster set C(f,p) is nowhere dense. Finally, in 1936, Jarnfk [ll] constructed a function having the 3-segment property at an un- countable set of points. Thus Bagemihl's theorem is not true for the 3-segment (hence 3-arc) property. As an initial response to Lohwater's proposal, Piranian [19], in 1959, proved that there exists a continuous function from D into W which possesses the 3-arc property at each point of C. Later that year Bagemihl, Piranian and Young [3] gave the following results on the 3-segment and 3-arc properties: There exists a function from D into W which has the 3-segment property at each point of C; there exists a continuous function from D into W which has the 3-segment property at each point of a perfect set; the elliptic modular function has the 3-arc property at every boundary point; and, there exists a Blaschke product which has the 3—arc pro- perty at each point of a perfect set on the unit circle. In the following year, the first of these four results was improved by Erdgs and Piranian [9] to read: There exists a function f in H for which each point p in R is the common endpoint of a family {L}p of rectilinear segments in H such that (1) {Llp contains 2 R0 elements, and the set of their directions is a set of second category and (ii) the intersection of the cluster sets of f along any three segments in [L]p is empty. A different type of result concerning the strong n-arc pro- perty was obtained in 1961. Young [22] proved that if f is a bounded analytic function from D into W which possesses the Z-arc property at a point p E C, then f possesses the strong n-arc property at p for all n > 2. Five years later, Mathews [17] established this result for arbitrary meromorphic functions and conjectured that if f is a meromorphic function from D into W which has the strong n-arc property at a point p E C, then f has the strong k-arc property at p for all k > n. In 1963, Rung [20, Theorem 5, p. 50] and Bagemihl [2, con- sequence of lemma, p. 4] independently established: If f is a normal meromorphic function from D into W, then for any integer n 2 2, the set of points at which f has the n-segment property is a set of first category and measure zero on C. It is still unknown as to whether the above theorem can or can not be stated for arbitrary meromorphic functions or at least for arbitrary holomorphic functions in D. To the author's know— ledge, the only progress toward this determination is attributed to McMillan [14, Theorem 4, po 10 and 15, Theorem 8, p. 195]. In his 1965 paper [14], he proved: If f is a holomorphic function from D into W and if f has no ambiguous points in the open are a of C, then the set of points at which f possesses the n-arc property for any n 2 2 is of first category on a. The following year in [15] he proved: If f is a holomorphic function from D into W and if a is an open arc of C such that the set of points at which f has a point asymptotic value is dense on a while f has the point asymptotic value m at no point of a, then the complement of the set of points at which f has the n-arc pro- perty for some n 2 2 is both metrically dense on a and residual on 0. Finally, in their 1966 paper, Bruckner and Goffman [5, Theorem 2, p. 512] proved: If f is a continuous function from H into R and if 51 and 52 are distinct directions in (O,fi), then for every point p E R, except for a set of first category, the supremum of the essential cluster set of f at p in the direction 31 is not exceeded by the infimum of the essential cluster set of f at p in the direction s2. The main purpose of this paper is to investigate the character of the sets of points at which continuous functions possess the n- segment property with various restrictions placed on the directions of a certain number of the segments involved. In section II we show among other things that the set of points p E R at which a con- tinuous function in H has the n-segment property relative to either n or n-l fixed directions is an F0 set of first category and measure zero. Then in section III we prove that the set of points p E R at which a continuous function has the n-segment pro- perty relative to n fixed points in H is again an F0 set of first category and measure zero. It is shown in section IV that the set of points p E R at which a continuous function has the essential—n-segment property relative to n fixed directions is of measure zero. In the last section we investigate the local behavior of a function f at points where f has one of the various are pro- perties. To mention one result, it is shown that even though a continuous function f in H may have the 3-segment property at a point p E R without p being an ambiguous point of f, each such point is in some sense ”very close" to being an ambiguous point if f is meromorphic. The remainder of this introduction will be used to indicate other notations and definitions along with an important theorem that will be used throughout this paper. OPEN SETS IN W. Let p(w1,w2) denote the (three-dimen- sional) Euclidean distance between the points W1 and W2 of W, let 8 be a countable basis for the topology induced by p on W, and let .& be the collection of all open sets expressible as a finite union of sets B E 3. Then for each integer n 2 2, let £?(n) be the set of subsets {Gl,...,Gn} of .& for which the intersection of all n of the sets G ,...,G is empty, where , l n J 01 is the closure of Gj in W. SEGMENTS IN H. If p is a point on R and if s is a real number in (0,fi), the set {2 E Z : arg(z ~ p) = s, 0 < Im(z) S l] is called a segment pp p and s is called the direction of this segment. The Symbols S(p) and Sj(p) (where j is a positive integer) will be used to denote various segments at p and the symbols dir S(p) and dir Sj(p), respectively, will denote their directions. The following comments pertain to the symbol S(p) but equally apply to the symbol Sj(p). The particular choice of the segment at p which S(p) is to represent will depend on the context in which the choice is made. In order to specify the seg- ment at p which S(p) is to represent, we need only specify its direction dir S(p). For example, if S(p) were chosen to rep- resent the vertical segment at p, we would say that S(p) is the segment at p for which dir S(p) = fi/2. For each real number r E (O,l], S(p,r) will denote the set S(p) 0 {z : 0 < Im(Z) 5 rl» Since every segment S(p) is an arc at p, the cluster set C(f,p,S(p)) of f at p along S(p) has already been defined. For convenience we shall use CfS(p), or simply CS(p) if f is clearly understood, to denote this cluster set. METRIC DENSITY AT A POINT. Let E be a Lebesgue measurable subset of R. If A is an interval of R containing the point p (which may not belong to E), the inferior and superior limits of m(E n A)/mA as mA approaches 0 are called the lower and ppper metric densities pf E 33 p respectively. If these are equal, their common value will be called the metric density pf E 35 p. If the metric density of E at p is 1, then we say p is a ppipg 2f density 9f E. In this paper, we will make considerable use of the following well-known theorem on density. LEBESGUE DENSITY THEOREM. f E is measurable, then the metric density pf E exists and ig egual Ep_l 33 every point pf E except for 3 set f measure zero. If the interval A is chosen so that p is the left (right) endpoint of A relative to an observer at z = -i, we can anal- ogously define the terms: lower and upper right (left) metric densities 2E. E at p; right (left) metric density pf E 35 p; .— and point 2f right (left) density pf E. II. THE n-SEGMENT PROPERTY RELATIVE TO k DIRECTIONS Bagemihl, Piranian and Young [3] have listed a number of open questions pertaining to intersections of cluster sets, one of which is the combination of the following two questions. QUESTION A. Does there exist a continuous function from D into W having the 3-segment property 35 each point 2£.E set pf positive measure 92 C? QUESTION B. Does there exist a continuous function from D into W having £22 3-segment property 3; SEER point 2£.§ §E£.2£ second category pp C? The attempt to answer these questions led to the results of parts 2 and 3 of this section. In part 2 we show that the answer to both questions is in the negative if two of the three directions of the segments are fixed. Then in part 3 we show that the answer to Question B is still in the negative if just one of the three directions is fixed. To make more explicit what is meant by fixing a certain number of the directions of the segments, we give the following definition. DEFINITION 1. Let f 23 a function from H into W, let n pg pp integer 2 2 and let k 23 pp integer satisfying 1 S k S n. Then f is said £9 possess the n-segment property g3 the point p 3p R relative 59 the k directions 5 ...,sk E (O,fi) if there 13 exist n segments Sl(p),...,Sn(p) 35 p with dir Sj(p) = sj for l S j S k such that the intersection of all n pf the sets 10 CSl(p),...,CSn(p) dig empty. l£>.ifl addition, the intersection pf any n—l 2f these sets is non-empty pg say that f possesses the strong n-segment property ap p relative £2.EEE k .EEXEE directions. NOTATION. For any set {sl,...,sk] of k 2 1 directions in (O,n), the set of points p in R at which f has the n-segment property (n 2 k) relative to these k fixed directions is denoted by EEn; 51,...,sk]. Furthermore, E[n] denotes the set of points at which f possesses the n-segment property. We remark that E[n; $1,...,sk] is a subset of EEn] and that E[n; 31,...,sk] is b f El: * “l f b I { * al} f a su set 0 n, 51,...,sm] or any Su set s1,...,sm o {sl,...,sk]. In part 1 it is shown that the sets EEn; 31,...,sk] and Efn] are measurable if f is a continuous function. In part 4, the set EEn] is shown to be countable for homeomorphisms of H onto a Jordan domain U. l. Measurability of the sets EEn; 51,...,sk] and EEn] LEMMA 1. at f pg a arbitrary function from H into W and let p _gqg point _f R. _f Sl(p),...,Sn(p) are segments 35 p for which the intersectiop pf all n 2; the sets CSl(p),...,CSn(p) * ii empty, then there exists 3 set {G1,...,Gn} E.& (n) and g positive rational number r S 1 such that f[Sj(p,r)] CIEj for l S j S n. Proof. Since each set CSj(p) is a closed subset of W, there exists an e > 0 such that the intersection of all n of the sets [C31(P)]€....,[C3n(p)]e is empty, where 11 [CSj(p)]e = {W E W : 9(W.[CSj(p)l) < e} . Due to the normality of W, there exist open sets U, J (l S j S n) such that cs. CU.CU.CCS. . Jo) J J [Jone Then, since each set CSj(p) is compact, there exist open sets Gj E A (1 s j s n) satisfying CSj(p) C Gj C Uj. It is clear that CSj(p) = fl EIEEIETI7EFT, where the inter- section is taken over all positive integers k. Each set ITEEYETI7R:IT]iS compact and a subset of fISEIBTI7RT]. Thus, for each j, there exists a positive integer k(j) such that WC Gj for all k > k(j). Choosing (l/r) = max {k(j) : 1 S j S n], we have f[Sj(p,r)] CIEj for 1 S j S n with Ej = ¢ and hence the "3:3 j l lemma is proved. In order to simplify to some degree the wording of the next result, we define sets 2*(n) as follows: Let 2 denote the set of closed intervals in (O,fi) which have rational endpoints. Then, for each positive integer n, let 2*(n) be the collection of all sets consisting of n mutually disjoint intervals in E, i.e. * {A1,...,An}€2 (n) if A162 (i=1,2,...,n) and AinAj=¢ for i f j. With the use of Lemma 1, the validity of the following de- composition is readily verified. 12 DECOMPOSITION A. Let f e a arbitrary function from H into W, et n pp; 33 integer 2 2 an 1e $1,...,sk pg any k (1 S k S n) fixed directions 33 (O,fl). or each set {A ’°""An-k} E 2*(n-k), each set {G ...,Gn] E £?(n) 53g each positive rational number r-S 1, define the set E[n; $1,...,Sk; A1’°°°’An~k; G1,...,Gn; r] pp pg the set pf points p 13 R pp which there exist n segments Sl(p),...,Sn(p) for which the following three conditions are satisfied: (1) dir Sj(p) = s, for 1 S j S k, J (2) dir Sj(p) E Aj-k for k+l S j S n, and (3) f[8j(p,r)] c:2% for 1 s j s n. Then EEn; s ..,s is the countable union of these sets. Likewise, 11" k ..._______._.__.._._.___.___ THEOREM 1. Let f be a continuous function from H into _ fl—mm— W, let n pp_§p integer 2 2, and let $1,...,S pg k (l S k S n) ____ -——— ——-e k fixed directions between 0 and n. Then EEn; 51,...,sk] and EEn] are Fo sets. 13 particular, the sets E[n; $1,...,sk; A1’°°°’An-k; G1,...,Gn; r] and EEn; A1,...,An; 01,...,Gn; r] are closed sets. 13 Proof. Let EEn; s A ..,A ,G ; r] be n-k; Gl’°°° n 1,...,Sk; 1,. an arbitrary set from Decomposition A and denote it simply by E. Suppose {pt} is a sequence of points in E which converges to a point p. We need only show that p E E. For each positive integer t, we have dir Sj(pt) = sj for l S j S k and dir Sj(pt) E A for k+l S j S n, where Sj(pt) j-k (1 S j S n) are the segments at pt guaranteed by pt E E. Choos- ing a Subsequence if necessary, we may assume lim dir S.(p ) = 3* tvw J t J for l S j S n. Then S? = S. for 1 S j S k and 5* E A, for J J J J-k k+1 S j S n. Let Sl(p),...,Sn(p) be the segments at p for which dir Sj(p) = s? for 1 S j S n. By the continuity of f, we have f[8j(p,r)] C-Ej for l S j S n. Hence p E E and E is closed. a If E is an arbitrary set in the decomposition of EEn], then a * slight modification of the above proof shows that E is closed and the theorem is proved. If Q is any countable set of directions between 0 and n, then, by taking the union of the sets E[n; 51,...,sk] over all integers n 2 2 and all subsets £51,...,s (1 S k S n) of Q, k} we obtain the following result. COROLLARY. Let f be a continuous function from H into W and let Q be a countable set of directions between 0 and n. Then the set of points 3p which f possesses the n-segment property for any n 2 2 relative pp some k (l S k S n) directions 13 Q is an F set. __ O'— l4 2. The cases k = n and k = n-l We will utilize the following lemma which is a consequence of a result of Doob [8, Lemma 2.1, p. 158]. LEMMA 2. Let E be a closed subset of the real line R and let p = 0 pp p point pp right density pp E. ‘_p h .32.3 continuously differentiable homeomorphism from p 2 0 into p 2 0 such that h(O) = 0 and h'(0+) > 0, then p = 0 pp p point pp right density of the set h(E fl R+). THEOREM 2. Let f be a continuous function from H into W and let n pp pp integer 2 2. Then the set of points pp which f possesses the n-segment property relative pp either n pp n-l fixed directions is pp first category and measure zero pp R. Proof. The result for n = 2 follows directly from Bagemihl's ambiguous point theorem. Thus we consider n 2 3 in what follows. Furthermore, since the result for n-l directions implies the result for n directions, we prove the former. Choose any n-l directions Sl’°"”’Sn—1 E (O,fi) and consider the set E[n; Sl’°°"’sn~l]° Pick an arbitrary set EEn; Sl’"°"’Sn-1; A1; G1,...,Gn; r] from Decomposition A and denote it by E. We now show that no point p E E is a point of density of E. Suppose q E E is a point of density of E. We may assume q = 0 and that dir Sn(0) < s, for 1 S j S n-l, where J 51(0),...,Sn(0) are the segments at 0 guaranteed by O E E. 15 For each j = 1,...,n-l define the function ”j from Ij = {p: p > 0. Sn(0) n Sj(p) aé as} into sn(0) by {ujm} = sum) n sj(p> Where dir sj(p) = sj. Then define the function hj from I, U {0] J into p 2 0 by hj(p) = luj(p)l for p > O and hj(0) = 0. It is easily verified that h (p) = (sin 8 /sin[s, — dir S (O) p . J J J n ])( ) By Lemma 2, the point 0 is a point of right density of hj(E fl Ij) for l S j S n-l. This implies that O is a point of n-l right density of O hj(E fl Ij). Therefore, there exists a positive J;l number b in this intersection sufficiently close to 0 such that if pj = p31(z), where z is the point on Sn(0,r) for which I2] = b, then 2 E Sj(pj,r) for 1 S j S n-l. Consequently _n~l z e s (0,1) m I n s.(p..r)l 1'1 '=1 J J J and, since 0 E E and p3. E E for 1 S j S n-l, we have f(z) E 61 O 62 fl...fl En which contradicts the fact that this inter- section is empty. Thus E contains none of its points of density. The Lebesgue Density Theorem gives mE = O and the arbi- tratiness of E gives mEEn; s ,s 1] = O in view of De- 1"‘° n— composition A. Furthermore, since E is closed (Theorem 1) and can not contain an interval (each interior point of which would be a point of density of E), EEn; s .,s ] is a first category 1’”” n-l set and the proof of the theorem is complete. 16 Remark. In [3] Bagemihl, Piranian and Young construct a con- tinuous function in H for which E[3; n/4, fi/Z, 3fi/4] contains a perfect set. Hence, Theorem 2 for n = 3 is the best possible re- sult in the sense that first category and measure zero can not be replaced by countable. The following corollary is established in the same manner as was the corollary to Theorem 1. COROLLARY. Let f pp_p continuous function from H into W and let Q pp ppy countable set of direcpions pp (0,n). Then the set pp points pp which f ppssesses the n-segment ppoperty for any n 2 2 relative to some set of n or n-l directions from Q is of first category and measppp zero on R. 3. The case k = l THEOREM 3. If f is a continuous function from H into W, then the set pp points pp which f possesspp ppphpppong n-segment property relative to ore fixed direction is of firsp category pp R. Proof. Choose any direction 5 E (0,fi) and let E be the set of points at which f possesses the strong n-segment property relative to s. If p E E, then there exist segments Sl(p),...,Sn(p) at p with dir 81(p) = s such that the intersection ofall n of the sets CSl(p),...,CSn(p) is empty while the intersection of the sets CSz(p),...,CSn(p) is non-empty. Therefore CSl(p) % C(f,p). Due to a result of Collingwood [ 6, Theorem 1, p. 5], the set of points p at which CS(p) % C(f.p) for dir S(p) = s is a set of first category. As a subset of this set, E is also a set of first l7 category and the theorem is proved. The following corollary is established by taking a countable union of sets considered in Theorem 3. COROLLARY. Let f be a continuous function from H into W and let Q pp p countable set pp directions pp (O,fi). Then the set pp points pp which f possesses the strong n-segment property for any n 2 2 relative to some direction in Q pp pp first category pp R. If a function f has the 3-segment property at a point p relative to a given direction 5, then either f has the strong 3- segment property at P relative to s or p is an ambiguous point of f. In view of this fact, the following theorem is an immediate consequence of Theorem 3. THEOREM 4. g; f is a continuous function from H into W, then the set of points pp which f possesses the 3-segment property 4. The n-segment property THEOREM 5. If f pp p homeomorphism pp the upper half plane H onto p Jordan domain U, then the set of points pp which f possesses the n—segment property for some n 2 2 pp countable. Proof. We first consider U = D and show that EEn] - EEn-l] is a subset of the set of ambiguous points of f. Suppose Sl(p),...,Sn(p) are n distinct segments at some point p E R for which the intersection of all n of the sets CSl(p),...,CSn(p) is empty while the intersection of any n-l of these sets is non-empty, and suppose that ' ' <...< ' . (2111' 51(1)) < dlr 82(1)) dir Sn(p) 18 = , d d Then CSl(p) fl CSn(p) 01 U dz, where a1 an a2 are close connected subsets of C with al N a2 = ¢ [note that one of the dj might be empty and one or both of them might be a single point]. Set A = {2; dir Sl(p) < arg(z-p) < dir Sn(p), 0 < Im(z) < 1}. If no point q E CSl(p) fl CSn(p) is accessible (see e.g. [7, p. 168]) from f(A), then for at least one of the sets aj, say a1, we have ¢ # al C CS(p) for all segments S(p) at p for which dir 31(p) s dir S(p) S dir Sn(p). In particular, al 0 such that pgplppy h E (0,Im(P)] 35g ppy a E (O,b) pp have m[Q[E(a),h,P] n A(a,h)] > [(n-1)/n][area A(a,h)] , where A(a,h) pp any triangular region pp H .pp height h whose base pp the interval (O,a) and which satisfies A(a,h) C Q[R(a),h,P]. Proof. If E' denotes the complement of E in R and if a so is a positive real number, then E'(a) = U Ij where the Ij are mutually j=1 disjoint open intervals of length 1*. Thus, for each h E (0,Im(P)], J we have (setting q = Im(P)) mEQEE'(a),h,P] n A(a.h)] s mQEE'] [ah/2n][2n(mE'(a)/a)]. Choose a real number b > 0 so that [mE'(a)/a] < (l/2n) for each a E (O,b) to get m[Q[E'(a),h,P] n A(a,h)] < (ah/2n) = (l/n)[area A(a,h)] which establishes the lemma. ,P E H Proof 2: Theorem 7. Choose any n points P1,... n and set d - min {Im(Pj): 1 S j S n}. For each positive real number 22 r < d and each set {Gl,...,Gn} in £fi(n), define EEGlyvu-,Gn; r] to be the set of points p E R for which f[8j(p,r)] Claj for 1 s j s n, where Sj(p) is the segment at p which passes through the point Pj for j = l,...,n. If E is the set of points at which f possesses the n- ,P , then segment property relative to the n fixed points P1,... n E = U E[G1,...,Gn; r] where the union is taken over all positive rational numbers r < d and all sets {G ,...,G j in £*(n). Since each of the sets 1 n E[Gl,...,Gn; r] is clearly a closed set, E is an F0 set. Pick an arbitrary one of the sets EEGl,...,Gn; r] and de- note it by E*. We now show that no point of E* is a point of density of E*. Suppose q E 3* is a point of density of E* and for convenience let q = 0° For each real number a > O, the intersection n n QER .Im(P.) ,PJ j=1 J J is an open triangular region A(a,ha) of height ha (0 < ha S d) whose base is the interval (0,a). It follows from Lemma 3 that for each j = 1,...,n there exists a real number bj > 0 such that for each real number a E (O’bj)’ we have V m[Q[E*(a),ha,Pj] fl A(a,ha)] > [(n—l)/n][area A(a,ha)]. If we set b = min {b1,...,bn}, then for each j = 1,...,n and for each real number a E (0,b) we have .4. 23 m[Q[E*(a),ha,Pj] n A(a,ha)] > [(n-l)/n][area A(a,ha)]. Consequently, for each a E (0,b), there exists a point 2a which is in each set Q[E*(a),ha,Pj] fl A(a,ha) for j = l,...,n. Hence, if we choose a point a E (0,b) for which ha < r, f(za) E 55 for each j = l,...,n. This contradicts {G1,...,Gn} E £*(n) and thus no point of E* is a point of density of E*. The Lebesgue Density Theorem gives mE* = 0 and thus mE = 0 by the arbitrariness of E*. Furthermore, since E* is closed and can not contain an interval (each interior point of which would be a point of density of E*), E is of first category and the proof is complete. Taking a countable union of the sets considered in the pre- vious theorem, we obtain the following result. COROLLARY. Let f be a continuOus function from H into W and let 9 pp p countable set pp points pp H. Then the set f points pp which f possesses the n-segpent property for some n 2 2 relative pp some n poinpg from -9 is an F set of first cate or o ___ __. ____5__Z and meaSure zero on R. IV. THE ESSENTIAL-n-SEGMENT PROPERTY RELATIVE TO n DIRECTIONS Let f be a continuous function from H into W, let S(p) be a segment at a point p E R and let mA denote the linear meaSure of A. A subset V of H is said to have positive upper density pp p relative pp S(p) if lim sup [m(V fl S(p,t))/m S(p,t)] > 0 t~0 The set V is said to have density d pp p relative pp. S(p) if lim [m(v n S(p,t))/m S(p,t)] = d ; tfio and, if d = l, p is said to be a point pp density pp V relative to S(p). With BBV,€] = pv' E W : ppv'pv) < e}, the essential cluster set, CeS(p), pp f pp p along S(p) is defined as follows: the point w E W is in CeS(p) if for every e > O, the set -1 . . f (BEW,€]) has pos1t1ve upper density at p relative to S(p). DEFINITION 3. f f is a continuous function from H into __ ____.___.__—_—.___——_._____ W, then f pp said pp possess the essential-n—segment property (n 2 2) 22 EEE RQEEE P E R relative pp n given directions ,sn E (O,fi) if the intersection of all n pp the sets __.—m S 1,”. cesl(p),ono.CeSn(p) 13 empty. where S.(p) (1 s j s n) is the J __.— segment pp p for which dir Sj(P) = S.« J The purpose of this section is to prove the following theorem. THEOREM 8. If f is a continuous function from H into W, then the set pp points p E R pp which f possesses the essential- 24 25 nesegpent property (n 2 2) relative pp n fixed directions pp_pp measure zero a We now prove three lemmas which will be used in the proof of Theorem 8. LEMMA 4. pp f is a continuous function from H into W, then CeS(p) is a closed set for each p E R and each segment S(p) at p. Proof. Suppose {wn} is a sequence of points in CeS(p) with wn H w. Let c > 0 be given. For some positive integer N = N(e), we have BEwN,e/2] C BEw,e]. Consequently -1 -1 f (BEWN.e/2]) C f (BEW.€]) -1 and hence the set f (BEw,e]) has positive upper density at p relative to S(p). Since a was arbitrary, w E CeS(p) and the proof is complete. LEMMA 5. Let f be a continuous function from H into W £121.93. 8(9) Eewgem 136R. I_f ceS(p)<:c for some open set G E.$, then p pp p point pp density pp f-1(G) relative pp S(p). Proof. If not, f-1(G') has positive upper density at p relative to S(p), where G' is the complement of G in W. Since G' n Ces(p) = ¢, to each w E G' there corresponds a real number 6(W) > 0 for which f-1(B[w,e(w)]) has density 0 at p relative to S(p). Then G' is contained in the union of the sets BEw,e(w)] taken over all w E G'. Since G' is compact, 26 k G' C U BEW.,€(W.)] for some finite subset {w1,...,wk} of G'. Thus -1 k -l f (G') C U f (BEW.,€(W.)]) j=1 J J which contradicts the assumption that f (G') has positive upper density at p relative to S(p) and the lemma is proved. Before stating Lemma 6, we introduce some notations that will be used throughout the remainder of this section. For any subset M of the real line R and any positive real number a, set M(a) = M H (-a,a). Then for any direction 5 E (0,”) and any positive real number h S 1, set T[M(a>.h.s] = U S(p.h) where the union is taken over all p E M(a) and where S(p) is the segment at p for which dir S(p) = s. LEMMA 6. Let E be a Borel subset of R and let s EE.§ direction pp (O,fi). For ppsitive numbers a and h (h s 1), let A(a,h) pp any open triangular region pp H pp height h whose base pp the interval (-a,a) and which satisfies A(a,h) C T[R(a),h,s]. f U pp pp open subset pp H for which there exists a real number q 6 (0,1) Such that m[U n S(p,h)] > q[m S(p,h)] 39; each p e E(a) where S(p) pp the segment pp p for which dir S(p) = s, then me O A(a,h)] > [q(mE(a)/a) - IJEarea A(a,h)]. Proof. If u is the characteristic function of E(a), then m 27 m(u n T[E(a>,h.s]> > (sin s) ffa qtms] u(p)dp Ii a j_a u(p)dp II (qh)mE(a) quE(a)/a][area A(a,h)]. Then, since [area A(a,h)] = (l/2)(area T[R(a),h,s]), we have mEU n A(a,h)] 2 m(U n T[R(a),h,s]) — [area A(a,h)] > q[mE(a)/a][area A(a,h)] - [area A(a,h)] as asserted in the lemma. Proof pp Theorem 8. Choose any n 2 2 directions 5 ,sn E (O,n) and let E be the set of points p E R at which 1,... f has the essential-n—segment property relative to these n directions1 If p is a point in E, then the intersection of the sets CeSl(p),...,CeSn(p) is empty, where Sj(p) (1 S j s n) is the segment at p for which dir Sj(p) = sj. Since the sets Cesj(p) are closed (Lemma 4), there exists a set £61,...,Gn} E £$(n) such that CeSj(p) CGj for each j = l,...,n. Furthermore, by Lemma 5, p is a point of density of f_1(Gj) relative to Sj(p) (1 S j S n). Hence, given an e (0 < e < l/Zn), there exists a rational number r > 0 such that m[f_1(Gj) n Sj(P,t)] > [(2n-1)/2n(l-e)]msj(p,t) for each j = l,...,n and each t E (0,r). 28 Let E[G1,...,Gn; r; e] be the set of points p E R such that mtf‘l n sj] > [<2n-1>/2n<1-e>]msj for each rational t E (O,r) and each j = l,...,n where Sj(p) is the segment at p for which dir Sj(P) = 8.. Then J E CIU E[G1,...,Gn; r; c] with the union being taken over all rationals r > 0, all rationals e E (0,1/2n) and all sets {G1,...,Gn} in £*(n). Each of the sets E[G1,...,Gn; r; e] is a G6 set since its complement can be expressed as the union of the closed sets —1 {p : mEf ] s [(2n-1)/2n<1-e>]msj(p,t>} taken over all rational numbers t E (O,r) and each j = l,...,n° Pick any one of the sets EEG1,...,Gn; r; e] and denote it by E*. To establish the theorem we need only show that mE* = 0. We now Show that no point of E* is a point of density of E*. Suppose p* E E* is a point of density of E*, and without loss of generality set p* = 0. Choose a real number a > O for which mE*(a) > 2a(l-e) and for which the intersection of the sets T[R(a),r,sl],...,T[R(a),r,sn] is a triangular region A of height h ([(2n-l)/2n(l-e)][2(l-e)] - l)[area A] = [(n-l)/n][area A] -l for each j = 1,2,...,n. Hence [f (Gj) H A] % ¢ in violation IIDCS j l 29 of the fact that {Gl,...,Gn} is an element in £?(n). Hence, no point of E* is a point of density of E*. Then, in view of the Lebesgue Density Theorem, we have mE* = 0 and the proof is complete. V. BEHAVIOR AT POINTS WHERE THE ARC PROPERTIES OCCUR The purpose of this section is to investigate the behavior of a function f at a point where f possesses either the n-arc or the n-segment property. We previously remarked (see p. 16) that continuous functions having the 3-segment property at an uncountable number of points do exist. Due to Bagemihl's ambiguous point theorem, we conclude that a point at which a continuous function f has the 3-segment pro- perty need not be an ambiguous point of f. However, if f is a meromorphic function, it is shown in Theorem 9 that such a point is either an ambiguous point of f or in some sense ”very close“ to being an ambiguous point of f. Combining Theorem 10 with the remark following it, we show that a holomorphic function can have the strong n—segment property at a point p for some integer n 2 2 and fail to have the strong k-segment property at p for any other integer k 2 2. Finally, in Theorem ll we show that if f is an arbitrary complex-valued function having the n-arc property at a point p for some integer n 2 2, then either p is an ambiguous point of f or the infimum of the diameters of all the arc cluster sets at p is positive. THEOREM 9. Let f pp p meromorphic function from H into W. _p f possesses the 3-segment property pp the point p in R, then there exist two disjoint Stoltz arcs 01 and 02 pp p for 30 31 which C(f,p,ol) fl C(f,p,oz) pp pp most p finite set pp points pp W. Remark. An are 0 at p E R is said to be a Stoltz are at p if there exist two distinct real numbers a and b in (0,fi) such that O'C {z E H : a S arg(z-p) S b}. Proof pp Theorem 9. For any point w' E W and any positive real number d S 1, let BEw',d] denote the set of all points w E W for which p(w',w) < d and let C[w',d] denote the set of all points w E W for which p(w',w) = d. Then let 8(f) denote the collection of all the ser BEw',d] whose boundary C[w’,d] contains none of r 1 the images of the points z E H for which f'(z) = O, and let ,&(f) denote the collection of all open sets G in W that are express- ible as a finite union of sets B E B(f). There exist three segments Sl(p), 82(p) and 33(p) at p with dir Sl(p) < dir 82(p) < dir S3(p), three open sets G G 3 1’ 2 and G3 in .&(f) with n G, = ¢, and a positive rational number i=1 3 r < 1 such that f(Sj(p,r) C Gj for j = 1,2,3. Furthermore, due to the fact that the boundary 5G of each G in .&(f) is the union of a finite number of circular arcs, we may also stipulate that 8G1 0 8G is at most a finite set of points in W. 3 For 1 S i < j S 3 and 0 < t S r, set A(i,j;r) = {z : dir Si(p) < arg(z-p) < dir Sj(p), 0 < Im(z) < r} and set £(i,j;t) = {z : dir Si(p) < arg(z-p) < dir Sj(p), Im(z) = t}. 32 Let A be the countable set of components A of the set {2 : z E A(l,2;r), f(z) E 6G1} for which I n 32(p,r) % ¢ (where I is the closure of A in H). Each component A E A is a homeomorphic image of the open unit interval 0 < x < 1. If A = ¢, then either f(52(p,r)) CIG or f(Sz(p,r)) C W - C l 1' In the former case, we have W _C = f c3 0 (cl n (:2) as and hence the desired Stoltz arcs at p are 01 = 32(p) and 02 = S3(p). In the latter case, the desired Stoltz arcs at p are 01 = Sl(p) and 02 = 82(p). Suppose A # ¢. If a component A E A has p as a limit point, then there exists an are a at p which is contained in A and C(f,p,a) C 6G1. Therefore C(f,p,a) n CSl(p) = ¢ and, in this case, 01 = a and 02 = Sl(p) are the desired Stoltz arcs at p. Suppose none of the k E A have p as a limit point. Since f is a local homeomorphism on [2 : f(z) E 3G1}, at most finitely many of the l E A intersect the line segment £(l,2;t) for any positive real number t S r. Furthermore, none of them intersect Sl(p,r). Consequently, there exists an are 01 at p contained in (s2 n {z : f 6 all) u (up where the second union is taken over all k E A. It follows that 33 In a similar manner we obtain an are 02 at p that is con- tained in (Sé(p.r) n {z : f e G,}> U (ux*) where the second union is taken over all components A* of the set {2 : z e A(2.3;rz f(z)_E 303} with i, n s2 # e. Hence Thus, in view of G. = ¢, we have j 1 J "DU-J C(f,p,01)fl C(f,p,dz) C 8G1 n 5G3 and the proof of the theorem is complete. THEOREM 10. There exists p holomorphic function f in H having the strong 3-segment property pp 0 which does not have the strong n-segment property pp 0 for any other integer n 2 2. Proof. Let 81(0), 82(0) and 33(0) be the segments at 0 3 with dir Sj(0) = jfi/4 for j = 1,2,3 and set U = U i=1 S,(0). J For positive integers n, define the sets Kn = {z : Im(z) = l/n, larg(z) - jfi/4l 2 l/n for j = 1,2,3] and R = {z : -n S Re(z) S n, l/n S Im(z) S n}. Also set R0 = ¢. Let be a continuous function from H into H which has go the strong 3-segment property at 0 relative to the directions fi/4, n/2 and 3fi/4. For each j = 1,2 and 3, choose an element 34 a, E C S.(0) n C S. J so J g0 J+1 By a well-known theorem of Mergelyan [18], there exists a (0) where we consider 84(0) = 81(0). polynomial P1(z) such that |P1(z) - g0(z)| < 1/2 on R3 n U and - < ' = . lPl(z) aj' 1/2 on R.3 fl Kj for j 1,2,3 Define g1(z) to be a continuous function in H such that 31(2) = P1(Z) on R3, gl(z) = g0(z) for Im(z) = 1/6 and z E U, and Ig1(z) - gO(z)l < 1/2 on U H [R6 - R3]. Suppose that we have defined continuous functions g0(z), gl(z),...,gn_2(z) in H and polynomials P1(z), P2(z),...,Pn_1(z) such that for each R = l,2,...,n-2 1) gk(2) = Pk(Z) on R3k . 2) gk(z) = g0(z) for Im(z) = l/(3k+3) and z E U , k 3) ng(z> - g0| < 1/2 on u n [R3k+3 - R3k] and for each k = l,2,...,n-l k 4) |Pk(z) - gk-l(z)l < 1/2 on R3k-3 U (U D R3k), and k _ < = 0°. _ 5) IPk(Z) ajl 1/2 on R3k fl K3q+j for q 0,1, ,k l and j = 1,2,3. Again by Mergelyan's theorem, if we define a continuous function on g (z) in H such that conditions 1), 2) and 3) are n-l 35 satisfied for k = n-l, there exists a polynomial Pn(z) such that conditions 4) and 5) are satisfied for k = n. Hence we can obtain a sequence [Pk(z)} of polynomials and a sequence {gk-l} of con- tinuous functions such that all five of the above conditions are satisfied for each k = 1,2,3,... It is readily verified that the sequence {Pk(z)} converges uniformly on each R3k and hence f(z) = pp: Pk(z) is a holomorphic function in H. Furthermore, |f(z) - g0(z)l H 0 as 2 H 0 with z e U and for each j = 1,2,3, [f(z) — aj| —. o as z e o with where the union is taken over all integers q = 0,1,2,3,... z E U K3q+j Then, since for each segment .S(O) at 0 with dir S(O) # jfi/4 (j = 1,2,3) there exists a positive integer N such that S(O) 0 Kn # ¢ for each n > N, f is theJdesired function. Remark. A construction similar to the one above enables us to show that given any integer n 2 2 there exists a holomorphic function in H having the strong n-segment property at inwhich does not have the strong k-segment property at O for any other integer k 2 2. If f is a function from H into W, the principal cluster set f f at p E R is defined to be the set E(f,P) = O C(f,P,0) where the intersection is taken over all arcs 0 at p. THEOREM 11. Let f pp pp arbitrary function from H into W. pp H(f,p) = ¢ for some point p E R, then either p pp pp ambiguous point f f pp there exists p positive number h such that the diameter pp C(f,p,o) pp greater than h for all arcs 0 pp p. 36 Proof. Suppose there exists a sequence {Gk} of arcs at p for which diameter [C(f,p,ok)] H 0. Without loss of generality, we may assume C(f,p,ok) H w for some w E w, Assume p is not an ambiguous point of f and let G be an are at p. Then, for each positive integer k, there exists a point wk 6 C(f,p,ck) fl C(f,p,OL Hence w E C(f,p,o) since C(f,p,o) is closed and wk 4 w. Since 0 was an arbitrary are at p, we have w E H(f,p) in violation of our hypothesis“ Thus p is an ambiguous point of f and the theorem is proveda BIBLIOGRAPHY 1. 10. ll. 12. 13. BIBLIOGRAPHY F. Bagemihl, Curvilinear cluster sets pg arbitrary functions, Proc. Nat. Acad. Sci. U.S.A. 41(1955), 379-382. F. Bagemihl, Some approximation theorems for normal functions, Ann. Acad. Sci. Fenn. Ser. A I 335(1963), 1-5. F Bagemihl, G. Piranian, and G. S. Young, Intersections of cluster sets, Bul. Inst Politehn Iasi (N. S. ) 5 (9), (1959), no. 3-4, 29-34. H. Blumberg, A theorem on arbitrary functions of two variables with applications, Fund— Math 16(1930), 17- 24— A. M. Bruckner and C Goffman, The boundary behavior pi real functions in theu upper half plane, Rev. Roumaine Math. Pures Appl. 11(1966), 507- 518. E.F. Collingwood, Cluster sets and prime ends, Ann. 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