ON THE LINEAR AND ANGULAR CLUSTER SETS OF FUNCTIONS MEROMORPHIC IN THE UNIT CIRCLE

ON THE LINEAR AND ANGULAR CLUSTER SETS OF FUNCTIONS MEROMORPHIC IN THE UNIT CIRCLE

BY

E . F . C O L L I N G W O O D

I.

I I . I I I . I V . V . V I . V I I . V I I I . I X .

Table o f Contents

Page

I n t r o d u c t i o n . . . 166

A g e n e r a l t h e o r e m o n c l u s t e r s e t s . . . 170

A t h e o r e m o n t h e c o v e r a g e o f r a d i a l c l u s t e r s e t s . . . ] 74

A p r o p e r t y o f t h e s e t I ( ] ) . . . 175

T h e s e t S (]) . . . 177

R e l a t i o n s b e t w e e n t i l e s e t s F ( / ) , I ( ] ) a n d S ( ] ) . . . 179

A t h e o r e m o n t h e m o d u l a r f u n c t i o n . . . 180

B e h a v i o u r in t i l e s e t of P l e s s n e r p o i n t s . . . 181

A t b e o r e m o n i s o l a t e d e s s e n t i a l s i n g u l a r i t i e s . . . 183

P o s t s c r i p t . . . 184

R e f e r e n c e s . . . 184

I n d e x o f s y m b o l s a n d t e c h n i c a l t e r m s Co (1, e~ o) ; CQ(~) (l, e ~ o) ; C A (1, e t o) . . . . w 1 C a ( / , e lO); C ( f , e tO); C C o ( / , e t~ e r e ; d e - g e n e r a t e , n o n - d e g e n e r a t e , t o t a l a n d s u b - t o t a l cluster sets . . . 2

F ( ] ) ; I ( [ ) ; F a t o u p o i n t ; F a t o u a r c ; P l e s s n e r p o i n t . . . 3

S (1) . . . 5

~ / ( 0 ) ; 7g(O) . . . w 0 c ~ o ) (l) ; s (0) . . . 7

w (/) ; W e i e r s t r a s s p o i n t . . . 8

SQ(r (/) ; Sa(o)(/) . . . 12

$ (l, 0) . . . 17

8~..~(r ) (]) . . . 18

S (9) . . . 19

166 E . F . COLLI~GWOOD

I. Introduction

I . L e t the function w = / ( z ) be m e r o m o r p h i c in the u n i t circle [z I < 1 a n d d e n o t e b y C~ = C o (/, e t o) the radial cluster set o/ / (z) at the point z = e i 0 which is defined as follows: a E C e ( / , e i~ if there is a sequence r l < r z < . . . < r n < . . . , l i m r ~ = l , such

n - - ~

t h a t l i m / ( r = e t~ = a . E v i d e n t l y Ce(], e t~ is a closed n o n - e m p t y set a n d is either a single point or a continuum. I n the f o r m e r case we s a y t h a t /(z) has a radial limit at the point z = e ~~

If we d o n o t e b y ~)(q) the chord of t h e u n i t circle t h r o u g h a p o i n t z = e ta a n d in- clined at the angle ~v, - ~ < ~v < ~, to the r a d i u s t h r o u g h e ~~ positive angles being measured to the right of the radius a n d negative angles to the left, we define the chordal cluster set C~(q~)(/, e t~ in a similar way. We s a y t h a t a EC~(~)(/,e t~ if there is a sequence t l > t 2 > " " > t n > " " , lim t = = 0 , such t h a t l i m / ( e t ~ 1 6 2 Again, Cq(r tO) is either a single point or a c o n t i n u u m , a n d Co(o)([,e t~ is the radial cluster set (:c, (/, et o).

Similarly, let A t)e an angle in the u n i t circle f o r m e d b y t w o chords passing t h r o u g h e t~ a n d define the corresponding angular cluster set Cz (/, e~~ W e s a y t h a t a E C , j ( / , e t~ if there is a sequence of points {z,} contained in A such t h a t lim z n = e t~

a n d l i r a / ( z , ) = a . Again, C d ( / , e ~~ is a closed n o n - e m p t y set a n d is either a single p o i n t or a continuum. 1 F u r t h e r , if zl x_= LJ 2 t h e n plainly Ca, (/, e ~~ =- Cz,(/, et~

2. We now define the o~tter angular cluster set 2 of /(z) at t h e point z = e t~ as t h e union

C~ (/, e ~ 0) = LI C~ (I, e ~ 0)

A

1 T h i s is t h e n o t a t i o n i n t r o d u c e d in COLLINGWOOD a n d CARTWRIGHT [ 5 ] , 1 ). 139. Generally, the n o t a t i o n we use is either t a k e n f r o m t h a t p a p e r , w h i c h will be cited h e r e a f t e r as C-C, or is derived f r o m it h~ an o b v i o u s w a y . A different c o n v e n t i o n h a s b e e n a d o p t e d b y J a p a n e s e a u t h o r s . ( N u m b e r s in b r a c k e t s refer to the list of references a t t h e e n d of the p r e s e n t p a p e r . )

T h e c o n n e c t i v i t y of CA a n d o t h e r c l u s t e r s e t s on locally c o n n e c t e d sets can be e s t a b l i s h e d b y ad hoc a r g u m e n t s (cf. C -C, pp. 90-92). T h e following general topological m e t h o d , w h i c h p u t s t h e m a t t e r in a few lines, I owe to P r o f e s s o r WILFRED KAPLAN. L e t Dt be t h e set of v a l u e s t a k e n b y ] (z) in t h e inter- section of ~ w i t h the d i s c e t 0 _ z < t. T h e n , w r i t i n g Dt for the closure of Dt, a n e q u i v a l e n t defini- tion of C A ( / , e t0) is CA(/,e tO)= N D t n, w h e r e tn'->O as n -->oo. Since Dt is c o n n e c t e d for all t it

n

follows b y a w e l l - k n o w n t h e o r e m (see, for e x a m p l e , I-IAUSDORI~F, Mengenlehre, p. 163, X V I I I ) t h a t CA (] e tO) is connected.

GROSS [6] u s e d the t e r m innere Hdu]ungsbereich for t h i s set.

C L U S T E R S E T S O F F U N C T I O N S M E R O M O R P H I C IN T H E U N I T C I R C L E 1 6 7

t a k e n over all angles A! b e t w e e n pairs of chords t h r o u g h z = e ~~ CA(/, e t~ is a n o n - e m p t y F . .

We c a n c l e a r l y define t h e cluster set of /(z) a t z = e ~~ w i t h respect t o a n y c u r v e i n Ix I<: 1 e n d i n g a t t h i s p o i n t , or a n y c o n t i n u u m or a n y sequence h a v i n g t h i s p o i n t as a f r o n t i e r or l i m i t p o i n t . B u t these d e f i n i t i o n s will be i n t r o d u c e d as t h e y are r e q u i r e d .

The cluster set C([, e ~~ of ](z) a t t h e p o i n t z = e ~~ is a f a m i l i a r n o t i o n 1 defined as follows: a E C ( [ , e ~~ if t h e r e is a sequence of p o i n t s {zn} c o n t a i n e d i n t h e u n i t circle such t h a t lim z n = e i~ a n d lira ] ( z ~ ) = a . C([, e i~ is e i t h e r a single p o i n t or a closed c o n t i n u u m .

W e shall say t h a t a cluster set or u n i o n of cluster sets is degenerate if it consists of a single p o i n t ; otherwise it is non-degenerate. 2 W e d e n o t e t h e c o m p l e m e n t s of ch,ster sets w i t h respect to t h e closed p l a n e or t h e sphere o n which t h e p l a n e is projected s t e r e o g r a p h i c a l l y b y C C 0, C Ca, C CA, "" etc. A cluster set or u n i o n of cluster sets whose c o m p l e m e n t is e m p t y , so t h a t it covers t h e whole w-plane or t h e whole w-sphere, we shall call total; a n d one whose c o m p l e m e n t is n o t e m p t y we shall call st~b-total, 3 t h e d e g e n e r a t e cluster sets b e i n g a sub-class of t h e s u b - t o t a l cluster sets.

3. T h e classical T h e o r e m o f F a t o u 4 s t a t e s t h a t i] ](z) is regulclr and bounded in

, < ~ - ~ [or

] z ] ~ . l then lira [(z) exists uni]ormly as z-~e ~~ in the angle ]arg ( 1 - z e - ~ ~ 2 all ~ > 0 and p.p. on the circum[erence I z I:: 1. T h a t is to say, [(z) has a n (outer) a n g u l a r limit, a n d therefore also a r a d i a l l i m i t p.p. I t was s u b s e q u e n t l y p r o v e d b y R. N e v a n l i n n a t h a t t h e s a m e p r o p e r t y holds i n t h e m o r e general case of a m e r o m o r p h i c f u n c t i o n [(z) whose c h a r a c t e r i s t i c T ( r , [) is b o u n d e d .

T h e c o m p a n i o n Theorem o] F and M . Riesz s t a t e s t h a t i[ !(z) is regular and ounded in I z I< 1 and has the same radial limit a, that is to say, i[ Ca([, e t ~ ]or a set o[ points e ~~ o[ positive measure on I z l = l, then [(z)F---a. This t h e o r e m was also e x t e n d e d b y R. N e v a n l i n n a to m e r o m o r p h i c f u n c t i o n s of b o u n d e d characteristic, s

Notation of C-C, p. 120.

2 Cf. WHYBURN, Analytic Topology, p. 30.

These notions, like that of the cluster sot itself, go back to PAINLEV~, C. R. 131 (1900), pp.

487-492, who spoke of singularities of a function as being points of complete or incomplete indeter- mination. We avoid the word "complete" because of its other uses in set theory

4 We use the notation p.p. to denote almost every.where; and m E or m (E) to denote the measure of a set E.

R . NEV~,NLIN~., Eindeutige analytlsche Funktionen (1936), pp. 190-197. Cited hereafter as

E.A.F.

168 ~. F. COLLINGWOOD

4. These f a m o u s t h e o r e m s h a v e b e e n d e e p e n e d a n d e x t e n d e d in a v a r i e t y of w a y s a n d h a v e g i v e n rise t o w h a t is n o w a n e x t e n s i v e t h e o r y . A n i m p o r t a n t gener- a l i s a t i o n of F a t o u ' s T h e o r e m for a n u n r e s t r i c t e d m e r o m o r p h i c f u n c t i o n is d u e t o P l e s s n e r . To s t a t e t h e t h e o r e m , a n d for s u b s e q u e n t d e v e l o p m e n t s , it is c o n v e n i e n t t o i n t r o d u c e some f u r t h e r d e f i n i t i o n s a n d t e c h n i c a l t e r m s . W e s h a l l call a p o i n t z = e ~~

a Fatou point /or ](z) i/ CA(l, e i~ is degenerate and i/ lim /(z) exists uni/ormly as z-~e i~ in any angle A between chords through e~~ a n d t h e set of F a t o u p o i n t s f o r / ( z ) on t h e c i r c u m f e r e n c e I z i = 1 we shall d e n o t e b y F (/). 1 W e shall call z = e t 0 a Plessner point /or /(z) i/ CA (/, e ~~ is total /or every angle A between pairs o/ chords through d o however small the angle may be; a n d t h e set of P l e s s n e r p o i n t s for /(z) on [ z l = 1 we s h a l l d e n o t e b y I(/). A n o t h e r n o t i o n t h a t will b e useful t o us is t h a t of a Fatou arc /or /(z). T h i s is d e f i n e d as an a r c of t h e c i r c u m f e r e n c e I z l = 1 w h i c h is a n open a r c of t h e f r o n t i e r of a s i m p l y c o n n e c t e d J o r d a n d o m a i n in ] z l < 1 in which e i t h e r [(z) or, for s o m e a ~ , 1 / ( / ( z ) - a ) is b o u n d e d . I t follows a t once f r o m F a t o u ' s T h e o r e m , b y c o n f o r m a l m a p p i n g , t h a t F ( / ) is p.p. on a Fatou arc. 2

T h e t h e o r e m of P l e s s n e r r e f e r r e d to a b o v e is

P l e s s n e r ' s T h e o r e m A.3 I / /(z) is meromorphic in ]z ] < 1, then almost all points o/

I z l = 1 belong either to F(]) or to I ( / ) .

W e n o t e t h a t I(]) ~ _ C F ( / ) . 4 I n d e e d , a P l e s s n e r p o i n t is in a n o b v i o u s sense t h e a n t i t h e s i s of a F a t o u p o i n t .

F o r a b o u n d e d f u n c t i o n , I ( [ ) is e m p t y a n d t h e t h e o r e m r e d u c e s t o F a t o n ' s T h e o r e m .

A s e c o n d t h e o r e m of P l e s s n e r g e n e r a l i s e s t h e Riesz T h e o r e m in a s i m i l a r w a y . T h i s t h e o r e m is

P l e s s n e r ' s T h e o r e m B. 5 I / /(z) is meromorphic in [ z [ < 1 and i/ ](z) has the same outer angular limit a, that is i[ C~(/, e~~ /or a set o/ points e t~ o/positive measure, then [ ( z ) ~ a .

i N o t a t i o n a n d terminology of C-C, p. 95.

2 Cf. C-C, p. 98.

a PLESSNER [10], p. 220. Plessner states his theorem in the slightly w e a k e r form w i t h the set (et 0} for w h i c h CA (f, e ~ 0) is degenerate in place of the set F (/) for w h i c h the angular limits CA (t, e ~ o) are also uniform. His argument, however, a c t u a l l y proves the theorem as w e state it here (arid as it is stated in C-C). It will be n o t e d t h a t C A (], e t 0) degenerate does n o t necessarily i m p l y t h a t lira ] (z) exists uniformly in every Stolz angle • at e ta as is the ease if e t e E F(]).

4 For sets on the circumference I z I = 1 the n o t a t i o n C of course denotes t h e c o m p l e m e n t s w i t h respect to t h a t space.

6 P L ~ S S ~ R [10], p. 224. This theorem w a s first proved for regular functions b y PRXVALOFF (see BI~BERBACH, Funktionentheorie, vol. II, 2nd ed., p. 158, or L v s I ~ a n d PRIVALOFF [8], p. 164).

C L U S T E R S E T S O F F U N C T I O N S M E R O M O R P H I C I N T H E U N I T C I R C L E 169 I t has b e e n s h o w n b y c o u n t e r e x a m p l e s 1 t h a t r a d i a l l i m i t s c a n n o t be s u b s t i t u t e d for o u t e r a n g u l a r l i m i t s in this t h e o r e m . To prove a u n i q u e n e s s t h e o r e m of this k i n d for r a d i a l l i m i t s a more s t r i n g e n t c o n d i t i o n m u s t be i m p o s e d on t h e set of p o i n t s e ~~

a t which t h e l i m i t is a t t a i n e d . L u s i n a n d P r i v a l o f f were t h e first to s h o w t h e signi- ficance of sets of category I I 2 i n t h i s p r o b l e m . I n order t o s t a t e t h e i r t h e o r e m we r e q u i r e t h e following d e f i n i t i o n . A set E is said to be metrically dense in (or on) an interval :r i/, given a n y sub-interval fl c ~, tile intersection E ~ fl is el positive measure.

F o r a n open i n t e r v a l t h i s is e q u i v a l e n t to s a y i n g t h a t E is m e t r i c a l l y dense a t e v e r y p o i n t of t h e i n t e r v a l . 3

W e c a n n o w s t a t e t h e

T h e o r e m of L u s i n a n d Privaloff, ~ I [ [(z) is regular in I z l < 1 and ha,~ the same radial limit a, i.e., i/ C~([,e~~ /or a set ~ ( 0 ) o/ points z = e ~~ ~ a ( O ) being both metrically dense and o] category I I on an arc ~ of ] z ] = l , then [ ( z ) ~ a .

More recently, in 1939, F. Wolf 5 p r o v e d a r e l a t e d t h e o r e m which we s t a t e in l a n g u a g e of cluster sets as follows:

W o i I ' s Theorem. Suppose that /(z) is regular in [z I < l and that there is a set

~ ( 0 ) of points z = e ~~ "re(O) being a Go dense on an arc Qr of [ z l = l , such that

~ E C Cq(/, e ~~ /or all e~~ E ~ ( O ) . Then i/ there is a number a r ~ such that aEC(,(/, e l~

p.p. on :~, we have ] ( z ) ~ a .

5. B y c o m b i n i n g t h e ideas in t h e proofs of these last two t h e o r e m s we prove a new t h e o r e m , no longer restricted to r e g u l a r f u n c t i o n s , which c o n t a i n s t h e m both.

T h i s is T h e o r e m 1. of t h e p r e s e n t paper. I n a d d i t i o n to t h e k n o w n t h e o r e m s of L u s i n a n d P r i v a l o f f a n d of Wolf, which emerge as corollaries, T h e o r e m 1 leads to n e w results o n t h e coverage of t h e r a d i a l cluster sets of /(z) o n a s u f f i c i e n t l y e x t e n - sive set of p o i n t s of t h e circumference I zl = 1. T h e t y p i c a l r e s u l t is T h e o r e m 2. T h i s in t u r n leads t o t h e i n t r o d u c t i o n of a n e w n o t i o n , n a m e l y t h a t of the set, which we denote by S (/), o/ points z = e t o at which the radial cluster set CQ (/, e t o) is total, a n d t h e i n v e s t i g a t i o n of its properties.

1 LusIN and PRIVALOFF [8], pp. 183-185.

2 Sets of the first and second categories are called of category I and of category I I respectively.

a Cf. HoBsoN, Theory o] Functions o/ a Real Variable, vol. I, 2nd ed., pp. 178-179, for the definition of a set metrically dense at a point. A set metrically dense on an interval is called by LusI~ and PRXVALOFF (I.c.p. 187) rdduit on that interval.

LusI~ and PRIV~.LOFF [8], pp. 187--1~8.

i F. WOLF [13].

1 7 0 E . F . COLLING'~V00D

T h i s set S(/) is t h e a n t i t h e s i s of t h e set a t w h i c h /(z) h a s a r a d i a l l i m i t , w h i c h l a t t e r is t h e set t h a t h a s h i t h e r t o b e e n m o s t closely s t u d i e d ; a n d t h e r e l a t i o n s h i p 1)etween t h e t w o sets m a v be e x p e c t e d t o be a n a l o g o u s t o t h a t b e t w e e n I ( / ) a n d F (/).

T h e r e is no s i m p l e r e l a t i o n b e t w e e n S(/) anti I(/), such as i n c l u s i o n of one in t h e o t h e r . W e p r o v e , however, t h a t S(/) a n d I ( / ) a r e t o p o l o g i c a l l y e q u i v a l e n t in t h e sense t h a t t h e y differ o n l y b y a set w h i c h is of c a t e g o r y I on t h e c i r c u m f e r e n c e I z l = 1 ( T h e o r e m 5) a n d t h a t if e i t h e r is dense on a n a r c :r of t h e c i r c u m f e r e n c e t h e n b o t h are r e s i d u a l on ~ ( T h e o r e m s 3, 6 a n d 7). W e t h u s o b t a i n e x i s t e n c e t h e o r e m s for S ( / ) a n d I ( / ) in t e r m s of one a n o t h e r a n d , in v i r t u e of P l e s s n e r ' s T h e o r e m A, in t e r m s also of t h e set F(/). W e can p r o v e , for e x a m p l e ( T h e o r e m 9), t h a t for t h e m o d u l a r f u n c t i o n /~(z) d e f i n e d in t h e circle I zl < 1, S(/e) is r e s i d u a l on l ul = 1.

A l t h o u g h o u r r e s u l t s a r e s t a t e d for t h e m o s t p ~ r t for r a d i a l c l u s t e r sets o u r m e t h o d s are e q u a l l y a p p l i c a b l e t o c h o r d a l o r m o r e g e n e r a l cluster sets, a n d t h e b a s i c l e m m a s are p r o v e d i H a f o r m s u f f i c i e n t l y g e n e r a l to y i e l d t h e s e e x t e n s i o n s of t h e m a i n results, t h e f o r m u l a t i o n s of which are o b v i o u s .

B y t h e s a m e n~etho(ts we p r o v e a s o m e w h a t s t r o n g e r f o r m of a r e c e n t t h e o r e m of ) I c i e r on t h e d i s t r i b u t i o n of t h e c h o r d s a t a given P l c s s u e r p o i n t on which t h e c h o r d a l c l u s t e r sets arc t o t a l . This r e s u l t , which is T h e o r c m 10, does n o t , howcver, e s t a b l i s h t h e e x i s t e n c e of a n y p o i n t z - - e t~ a t w h i c h t h e r a d i a l c l u s t e r set or a c h o r d a l cluster set iu a givc,l d i r c c t i o n is t o t a l . I t is T h e o r e m 6 t h a t is t h e e x i s t e n c e t h e o r e m for such points.

F i n a l l y , for c o m p l e t e n e s s , a n d a l t h o u g h it does n o t a c c o r d s t r i c t l y w i t h t h e t i t l e

~)f t h i s p a p e r , we give a t h e o r e m on t h e d i s t r i t ) u t i o n of t h e t o t a l l i n e a r c l u s t e r sets of ~t uniform f u n c t i o n a t a u isolate(l e s s e n t i a l s i n g u l a r i t y . T h i s is T h e o r e m I I . I t is an o b v i o u s a n a l o g u e of T h e o r e m l{) a n d is provc(l in t h e s a m e w a y .

II. A general t h e o r e m on linear cluster sets

6. W e begin b y p r o v i n g as a l e m m a w h a t a m o u n t s to a g e n e r a l i s a t i o n to m c r o - m o r p h i c f u n c t i o n s a n d to linear, b u t n o t n e c e s s a r i l y r a d i a l , c l u s t e r sets of t h e p r e - l i m i u a r y p a r t of W o l f ' s t h e o r e m . T h e p r o o f d e r i v e s p a r t l y f r o m W o l f a n d p a r t l y from Lusin a n d Privaloff. W e shall use 7~/(0) anti 7/(0) t h r o u g h o u t t o d e n o t e sets of p o i n t s e t~ on t h e c i r c u m f e r e n c e l zl = 1.

L e m m a 1. Suppose that ](z) is meromorphic in I zl < 1 and that /or some ]ixed

~, - 2 <q J < 2 , and some complex number a, /inite or in]inite, there is a set ~ ( 0 ) o/

C L U S T E R S E T S O F F U N C T I O N S M E R O M O R P H I C I N T t I E U N I T C I R C L E 171 points z = e ~~ o/ category I I o n a certain arc ~ o/ the circum/erence [z I = 1 and such that a E C C e ( r ~~ /or all e * ~ Then the arc cr contains an arc fl such that (i) fl is a Fatou arc /or [(z) in the neighbourhood o/ which either [(z) or 1 / ( [ ( z ) - a ) is uni- /ormly bounded according as a = ~ or a # ~ , and (ii) ~ ( 0 ) is dense on ft.

I t is sufficient to prove t h e l e m m a for t h e case a = ~ since otherwise we have o n l y t o m a k e a linear t r a n s f o r m a t i o n of /(z) which carries a into ~ .

D e n o t e b y A , ( T , N, O) t h e set of points e *~ 0_<0_<2,~, such t h a t , for all values of t in 0 < t < T ,

I/(e~~ - t e ~ ) ) l < N .

T~ > T2 > ... > T~ > ... , lira T ~ = 0 . N o w take

T h e n

Also, for a n y ~/> 0,

A , (T~, N, O)~_A,(T,,+1, N , 0).

A r N , O ) c _ A r N - § O) so t h a t if we t a k e N 1 < N 2 < - - - < N~ < . . . , lim N, = ~ , we have

while plainly

A , = A ~ ( T ~ , N~, O)c_Ar247 N,,+~, 0)--A,,~t, 7~ (o) = ( O A,,) n or

v

(l)

Since, b y hypothesis, ~ ( 0 ) is of c a t e g o r y I I in ~ it follows t h a t at least one of the sets A,,(1 a, say Ak(1 :r is of c a t e g o r y I I ill :r There is therefore an arc fl___:(

such t h a t Ak is (lense (m /~; and, since A ~ _ ~ ( O ) , the set ~ ( 0 ) is <lense on ft.

F o r e ~ ~ a n d for all 0 < t < T k we have

I/(e~~ 1 - t e'~)) I < N~ ; (2)

and, since Ak is dense on fl, it follows from this t h a t the inequality

Jl(z)i<_w~

(3)

is satisfied t h r o u g h o u t t h e a n n u l a r quadrilateral B (not containing t h e o r i g i n ) d e f i n e d b y the arc fl, the two chords at the end points of fl inclined at the angle q to the respective radii at these points, a n d the circular arc [ z l = 1 - Tg cos q joining the two chords. F o r e v e r y interior p o i n t of B is, b y (2) a n d the fact t h a t A~ is dense on fl, a limit of points at which [ / ( z ) ] < N k , so t h a t /(z) can h a v e no poles in B a n d is therefore continuous at e v e r y interior point of B, a n d (3) is therefore satisfied a t e v e r y such point. This completes the proof of the lemma.

172 E. F. COLLINGWOOD

F o r ~ = 0 t h e chord 9(q~) t h r o u g h e ~~ becomes t h e radius at this p o i n t a n d Ce(o)=Ce(/,e i~ is t h e radial cluster set at z = e ~~ Since in t h e l e m m a radial a n d chordal cluster sets are on exactly the same footing we shall as a rule limit the later applications of the results of this section to radial cluster sets, except where t h e chordal cluster sets are necessarily involved in t h e s t a t e m e n t of a theorem, leaving the corresponding generalisation for chordal cluster sets t o be understood.

As a first application of L e m m a 1 we o b t a i n our general theorem, n a m e l y

7g

Theorem 1. I / /(z) is meromorphic in [z[ < 1 and i/, /or a constant q~, - ~ < q~ < 2 , there is a set ~ ( 0 ) o/ points z = e ~~ o/ category I I on an arc :r o/ the circum/erence ] z [ - 1 such that the intersection [1 CC0(~)(/, e ~~ is not empty and i/, /urther, there

ei 0 ~ ?M (0)

is a number b, /inite or in/inite, and a set ~ ( 0 ) metrically dense on ~ such that bE n Co(,t~((/,et~ then / ( z ) ~ b .

e t 0 ~ }l(O)

B y L e m m a 1, :r contains an arc /~ which is a F a t o u arc f o r / ( z ) . Therefore, b y F a t o u ' s theorem, F ( J ) is p.p. on fl a n d so, since m ( ~ l ( 0 ) N f l ) > 0 b y hypothesis, it follows t h a t

(m ( ~ (0) n F (1)) ~: m ( ~ (0) n fln F (1)) > o.

Now, for e t~ E F (/) we h a v e CA (/, e t~ = Cr (f, e t~ and, b y hypothesis, Co(q, ) ([, e ~ o) = b for el~ E ~ (0) N F (1). Since m (TI (0) N F (f)) > 0 it now follows from Plessner's T h e o r e m B t h a t / ( z ) - : b a n d the t h e o r e m is proved.

Corollary 1. I / /(z) is meromorphic in I z l < 1 and there is a number b and a set

~ b ( O ) which is both metrically dense and o/ category I I on an arc cr o~ the circum- ]erence I z I = 1, such that the radial cluster set C o (f, d o) = b for all e t o ~ ~lb (b), then f (z)---- b.

This is immediate. W e have only to p u t (0) = ~ (0) -- ~ (0).

This corollary extends to m e r o m o r p h i c functions t h e t h e o r e m of Lusin a n d Privaloff q u o t e d in the I n t r o d u c t i o n (w above). These a u t h o r s also c o n s t r u c t e d counter examples which show t h a t their t h e o r e m is best possible in t h e sense that, given the condition t h a t ~ ( 0 ) is of c a t e g o r y I I on ~, the condition t h a t it m u s t also be metrically dense on cr c a n n o t be dispensed with, a n d conversely. 1 T h e o r e m 1 m u s t therefore also be best possible in a similar sense.

1 LusIN and PRIVALOFF [ 8 ] , pp. 185-186.

C L U S T E R S E T S O F F U N C T I O N S M E R O M O R P I I I C I N T t I E U N I T C I R C L E 173 7. We conclude this section with a r e m a r k on the extension of L e m m a 1 a n d its consequenses to more general linear cluster sets. Consider a curve 2 in I zl < 1 tending to the circumference I z l = 1. T h e " e n d " of 2 m a y be either a p o i n t or an arc of I zl = 1 or, as in the case of a spiral, the whole circumference. F o r simplicity we m a y assume t h a t ). has only one point of intersection with e v e r y circle I z I = r < 1 so that, on 2, I zl is strictly increasing as z tends to the circumference ] z l - 1 . To define the orientation of 2 we m a y t a k e the a r g u m e n t 00 of a n y point z 0 on 2 other t h a n the origin. B y r o t a t i o n a b o u t the origin t h r o u g h an angle 0 - 0 0 we o b t a i n the family {2(0)} of r o t a t i o n a l t r a n s f o r m s of 2 = 2 ( 0 0 ) , the original curve. E v i d e n t l y , as 0 sweeps o u t an interval the curve 2(0) sweeps o u t a d o m a i n in I z [ < l whose frontier includes an arc of the circumference I z [ = 1 which m a y be wholly or p a r t l y inaccessible f r o m the domain. T h e cluster set C~(o)(]) of /(z) on a curve )L(0)is defined as follows: a E C~(0)(/) if there is a sequence of points (z~} on ,~ (0) such t h a t lim I z~ I = 1 a n d l i m / ( z n ) = a . C~(o)(/) is again a n o n - e m p t y closed set which is a c o n t i n u u m if

n - - ~ o o

it is non-degenerate. F o r a given family of curves (~(0)} the result corresponding to L e m m a 1 is

L e m m a 1 a. Suppose that /(z) is meromorphic in I zl < 1 and that, [or a given /amily o/ curves {~(0)~, which are rotational trans/orms o/ om~ another and on w h i ~

I zl is strictly increasing as z tends to the circum/erence [zl:: l, there is a complex number a, /inite or in/inite, and a set F~(O) o/ v~ints 0 o/ category I I in an interwd

~r ~ (0, 2~) such that

aECC~(o)(/) /or

Again, there is no loss of generality in t a k i n g a = ~ . F o r a given value of 0 every point of ).(0) is d e t e r m i n e d b y I zl, z E).(0). D e n o t e b y L ( T , N, 0) the set of

- - ~ Z ~

values of 0 , 0 ~ 0 ~ 2 ~ , such that, for all values of zE).(0), where 1 T I,] 1, the inequality I/(z) l < N is satisfied, T a k i n g T I > T 2 > . . . > T v > . . . , lim T ~ = 0 , and N 1 < N 2< ... < N~ < . . . , lim N ~ = ~ , we have, as before,

r - ~ o o

a n d

L~=L(T~, N~, O)~_L(T~+I, Nv+l, O)=L,,+1 s = ( U Lv) N a.

v

Since s is, b y hypothesis, of c a t e g o r y I I in :~, at least one of t h e sets Lk N a is

174 E. F. COLLINGWOOD

of category l I in :r a n d is therefore dense in a sub-interval fl_~a. The proof n o w proceeds e x a c t l y as for L e m m a 1 a n d need n o t be repeated.

The result can easily he generalized b y rclaxing the restriction t h a t I z], z E).(0), is one valued for a given 0.

I l L A t h e o r e m on the c o v e r a g e o f the radial cluster sets 8. As a f u r t h e r consequence of T h e o r e m 1 we prove

Theorem 2. I[ /(z) is meromorphic in I z l < 1 and i/ there is a set ~ ( 0 ) o / points z = e '~ metrically dense on an arc ~ o/ the circum/erence [zl = 1 such that N Ce([, e ~~

is not empty, then either given any set ~ ( 0 ) o/ category I I on ~, the union LJ C~([, e ~~

m (0)

is total, or /(z) is a constant.

Suppose t h a t I,J Ce(/, e ~~ is s u b - t o t a l so t h a t its c o m p l e m e n t I"1 C CQ(/, e ~~ is

m (0) r~ (0)

n o t e m p t y . Since ~ ( 0 ) is of c a t e g o r y I I on a the condition of T h e o r e m 1 is satis- fied a n d it follows t h a t if there is a n u m b e r bE N Co(/,el~ t h e n / ( z ) ~ b . This proves

(O)

the theorem.

W e denote b y W(/) the set of (Weierstrass) points z = e Is for/(z), i.e. the points e ~~ for which C (/, e t~ is total. 1 T h e o r e m 2 supplements the following quite trivial ot)servation, n a m e l y , t h a t i/, /or a set ~ (0) metrically dense on ~, N C~([, e t~ is not

~(0)

empty, then ~ - W ( ] ) . F o r if e t ~ W(/) so t h a t C C ( / , e ~~ is n o t e m p t y , then e ~~

is contained in a F a t o u arc fl; an(l, for e~~ ~ ( 0 ) , we can t a k e f i g : r B u t , b y Pless- net's T h e o r e m B, we m u s t have m ( ) l ( 0 ) N f l ) = 0 ; for otherwise there is a subset of F ( / ) N fl of positive measure in which the angular limit of /(z) is constant, so t h a t [(z) is a constant. This contradicts the h y p o t h e s i s t h a t ~ ( 0 ) is metrically dense on :r and our assertion is proved.

An i m m e d i a t e corollary of T h e o r e m 2 is

Corollary 2, I / / (z) is meromorphic in I z [ < 1 and i / a E Cq (/, e ~ o) /or all e ~ ~ E ~ (0), where ~ ( 0 ) is metrically dense on an arc ~ o] [z I= 1, then either C ~ ( O ) is residual on r162 and [.J Ce(/, e t~ is total, or / ( z ) ~ a .

C~(O)

For, if ~ ( 0 ) is of c a t e g o r y I I on ~, t h e n / ( z ) ~ a , b y Corollary 1.

This corollary is illustrated b y a n u m b e r of k n o w n examples, as for instance

i C - C , p. 137.

CLUSTER SETS OF FUNCTIOIgS MEROMOR:PHIC 1N THE U N I T CIRCLE 175 K o e n i g s ' f u n c t i o n K ( z ) l , a f u n c t i o n co(z) c o n s t r u c t e d b y L u s i n a n d P r i v a l o f f 2 a n d a f u n c t i o n /(z) r e c e n t l y i n v e s t i g a t e d b y B a g e m i h l , E r d 6 s a n d Seidel. a I n all t h e s e cases t h e r e is a r a d i a l l i m i t a = C o ( / , e ~~ in a set T/(0) which is p.p. on t h e c i r c u m f e r e n c e ; a n d it is e a s i l y verified t h a t t h e c o m p l e m e n t a r y set C ~ (0) is in e a c h case r e s i d u a l on t h e circumference.

IV. A property o f the set

l ( f )

9. W e shall p r o v e t h a t if t h e set I ( / ) of P l e s s n e r p o i n t s of /(z) is dense on an a r c :r t h e n it is also r e s i d u a l on t h a t arc. To do t h i s we r e q u i r e t w o f u r t h e r l e m m a s t o s u p p l e m e n t L e m m a 1. F i r s t we p r o v e

I /

/or

- < ~ < 2

L e m m a 2. l(z) is meromorphic in I z l < 1 and il, fixed 2

there is a set ~1(0) o~ category I I o u an arc ~ o I the circum/erence I z l = 1 such that Co(q,)(/,e ~~ is sub-total /or all e*~ ETtl(O), then there is a sub-set ~/0(0)_~]ff/(0), also o/

category I[ on a, such that ['1 C Co(r e*~ is not empty.

~o 0

I f (~,(~,)(/, e ~~ is s u b - t o t a l we can f i n d a circle c o n t a i n e d in C (',,(q.)(/, e i~ since t h i s is a n ()pen set.

N o w s u p p o s e t h e w - p l a n e t o I)e p r o j e c t e d s t e r e o g r a l ) h i c a l l y o n t o t h e u n i t W-sl)here so t h a t a circle in t h e p h m e is p r o j e c t e d o n t o a circle on t h e sphere. A d a p t i n g P l e s s n e r ' s m e t h o d , we c o n s t r u c t on t h e s p h e r e a sequence of finite t r i a n g u l a r l a t t i c e s It, l.z, ...I,,, ... etc., e a c h l a t t i c e heing a s u b - d i v i s i o n of its I)redecessor a n d such t h a t t h e l e n g t h of t h e longest side of t h e l a t t i c e /~ is less t h a n 2 " , say. W e d e n o t e t h e i n d i v i d u a l t r i a n g l e s in t h e l a t t i c e 1, b y d,.~, d,.,. . . d,.r,(,). D e n o t e b y 1',,(0)the set of p o i n t s e ' ~ such t h a t n is t h e s m a l l e s t n u m b e r for which CU~(,~(I,e t~ con- t a i n s t h e whole of a t least one of t h e t r i a n g l e s d,,,., 1 ~ v - ~ m ( n ) . T h e n e v i d e n t l y

I l l (0) _c/'2 (0) c . . . c 1"~ (0) ~ - . . . a n d

m ( 0 ) = U V ~ (0).

n

We now s u b - d i v i d e t h e sets /'~ (0) as follows. W e assign to each t r i a n g l e d~. 1, d,.2 . . . . d,. ,,(,)

all t h o s e v a l u e s of e ' ~ (0) for which d ... l < v < m (n), is c o n t a i n e d in C Cocr (/, el~

t C-C, p. 94.

s LUSlN and PRIVALOFF [8], p. 189.

a BAOEMIHL, ERD•S and S~ID~L [1], p. 139 ; also J. WOLFF [14] and [15].

1 2 - 533807. A c t a M a t h e m a t i c s . 91. Imprim6 le 28 octobre 1954

176 E . F . COLLI~GWOOD

and we denote the corresponding sub-sets of l ~. (0) b y ~) .... (0), 1 _< ~_< m (n). All b u t one of these sub-sets :O .... (0) m a y be e m p t y , b u t ~t least one of t h e m m u s t be non- e m p t y , and a n y two of t h e m which are n o n - e m p t y m a y h a v e c o m m o n points. W i t h these definitions we e v i d e n t l y h,~ve

?fl(0)=U U O,,~(0). (4)

Since ~ ( 0 ) is of c a t e g o r y I I on ~ one at least of the enumcrable set of sets

~ , . ~ ( 0 ) , n < ~ , v ~ m ( n ) , s a y :Dj.k(0), m u s t be of c a t e g o r y I I on ~. But, for all ei~ the triangle dj.k is contained in C C q ( o ( I , e i~ and so, p u t t i n g ~ 0 ( 0 ) =

= ~i. ~ (0), the l e m m a is proved.

lO. Secondly, We prove

L e m m a 3. I / /(z) is meromorphic i~ I zl < l and i[ there is a set ~ ( 0 ) o/ points z = e i~ satis[ying the conditions (a) that ~ (0) ~_ C I (/), and (b) tt, at ~ (0) is o/ cateqory ] I on an arc o~ o/ the circum/erenee t zl==], then there is an arc f l ~ : r such that ( i ) / ~ i s a F~lto~ arc /or /(z), and (ii) ~ ( 0 ) is dense o~ ft.

F v c r y point ~ ~ C I ( / ) is the vertex of an angle A (0) iu [z I-: 1 in which the angular cluster set C,t(o)(1, e ~~ is sub-total. N o w d e n o t e b y E 1 the subset of ~ ( 0 ) at each point e ~~ of which there is a A (0) of m a g n i t u d e greater t h a n ~z/2 such t h a t ('~(0)([,e ~~ is sub-total, b y E z the subset of ~ ( 0 ) at which there is ~ A ( O ) o f m a g n i t u d e greater t h a n ~,'4, h y En the subset of ~ ( 0 ) at which there is a A ( 0 ) o f m a g n i t u d e greater t h a n ~ / 2 " in which C',~(o)([,e ~~ is sub-total, an(l so on for inde- finitely increasing n. Evi(|ently E t g E., ~ _ ... ~_ E , ~_ ..., a n d

771 (0)= U E..

n

We now proceed to sub-divide each set E , into a finite set of subsets as follows.

(1 - 2 - " ) ~

")

f)ividc the angle -- '2 "~ q) < 2 (1 - 2 on the interior n o r m a l side of the t a n g e n t to the unit circle at e i~ into N ~ = 2 n - 1 equal parts of m a g n i t u d e ~ / 2 n b y drawing the chords ~)t =~(q~l), ~2~=ff(q~.~) .. . . ~2u=~)(q~,v) t h r o u g h e ~~ T h e n e v e r y angle A(0) of m a g n i t u d e greater t h a n re/'2 ~ a n d contained in ] z l < 1 m u s t contain at least one of t h e chords ffl, 02, ..-flu. F o r otherwise it m u s t contain one of t h e half-tangents at e f~

c o n t r a r y to the definition of A(0). We denote b y E .... l _ ~ v _ < N , t h e subset of En for which there is a A(0) of m a g n i t u d e greater t h a n ~z/2" such t h a t C~(o)(f,e i~ is

C L U S T E R S E T S O F F U N C T I O N S M E R O M O R P H I C I N T H E U N I T C I R C L E 1 7 7

s u b - t o t a l a n d which contains e," Clearly, a point e ~~ m a y belong to more t h a n one of the sets En,~. W i t h these definitions we h a v e

( 0 ) = U U E .... (5)

n l ~ v ~ N

Since ~ ( 0 ) is of c a t e g o r y l I on ~ one at least of t h e sets E ... say Ej,k, m u s t be of category I I on ~. Consequently, for all e ~~ E Ej, k, Cr k (/, e~~ which is contained ill a sub-total set C,J(o)(/, e~~ is s u b - t o t a l ; a n d it follows f r o m L e m m a 2 t h a t there is a subset )~0 (0)_c Ej, k--~ ~ (0), also of c a t e g o r y I I on :r such t h a t n c c,, k (/, e ~~

'm~ (0)

is n o t e m p t y . I t n o w follows f r o m L e m m a 1 t h a t there is an arc f i g : r such t h a t (i) fl is a F a t o u arc for /(z), a n d (ii) ~ 0 ( 0 ) , a n d therefore also ~ ( 0 ) , is dense on ft.

The l e m m a is therefore proved.

l i . We are n o w in a position to prove

Theorem 3. I / /(z) is meromorphie in I z l < 1 and the set I([) o/ Plcssner points o[ / (z) is dense on an are :r o/ the circum/erence ] z [ ~-= l, then I (/) is also residual ou ~.

Suppose t h a t C I ( / ) is of c a t e g o r y ] I on ~. Then, p u t t i n g ~ ( 0 ) : = C I ( / ) a n d a p p l y i n g L e m m a 3, it follows t h a t there is a F a t o u arc f l _ ~ . But, since evidently f l ~ - C I ( / ) , this implies t h a t I ( / ) is n o t dense on :r and the theorem is proved.

As a further consequcncc of the a r g u m e n t we have

Theorem 4. I[ [(z) is mcromorp)~ic in [ z [ - : 1 elnd i/ on an r :r o/ the circum.

/er~nce [z[=:l, m ( F ( ] ) N a ) = O , then F ( / ) is o[ cat~gory I o , ~.

F o r F ( / ) ~ _ C I ( / ) so t h a t C I ( / ) N : r is of category I I if F ( / ) N ~ is of category H ; and this implies t h a t ~ contains a F a t o u arc/~. Since m (F (f) N :r • m (F (/) N/~) > 0 this proves the theorem.

V. The set

S(f)

12. The set S(/) is defined as the set of points z = e ~~ for each of which the radial cluster set C~,(/, e t~ of /(z) is total. The definition can obviously be e x t e n d e d to chordal or more general linear cluster sets, as for example the set Se(~) of points e t~ for each of which Cr e~ t~ is total, or t h e set S~(o)(/) on which C~(o)(f, e t~ is total. We shall, however, confine t h e detailed discussion t o t h e radial case S ( / ) since it will be obvious t h a t o u r theorems are equally applicable to Sq(~) (/) and, u n d e r a p p r o p r i a t e limitations on ~(0), to S~(o)(/) also. We first prove

1 7 8 E . F . COLLINGWOOD

Theorem 5. I[ /(z) is meromorphic in ] z t < 1, t/ten the sets S ( / ) and I(]) di//er by a set o[ category I on the eircum/erence I zl = 1.

We prove first t h a t S ( / ) n C I ( / ) is of c a t e g o r y I. To (lo this, p u t ~ ( 0 ) =

= S (f) n C I (/) and apply L e m m a 3 under the hypothesis t h a t ~ / ( 0 ) is of c a t e g o r y I I . This implies the existence of a F a t o u arc fl on which ~ (0) is (lense. B u t evidently no point of fi can l)etong to S ( / ) so t h a t f l ~ - C S ( / ) and ~ ( O ) N fl is e m p t y . This contradiction proves our ~ssertion.

Similarly, we prove t h a t C S (/) N I (/) is of category I. P u t 771 (0) -- C S (/) f / I (/) a n d a p p l y L e m m a 2 under the h y p o t h e s i s t h a t ~ (0) is of c a t e g o r y I I . This implies the existence of a subset ~ 0 ( 0 ) ~ _ ~ (0) such t h a t n c c ( , (/, e is) is n o t e m p t y . Now,

"me (0)

applying L e m m a 1, it follows t h a t there is a F a t o u arc fl on which ~ 0 (0), a u d there- fore also ~ (0), is dense. B u t evidently no point of fl can belong to I ( / ) , so t h a t f l ~ C I ( / ) an(1 we again have a contradiction, since ~ (0)f'l fl m u s t be e m p t y . The t h e o r e m is therefore provc(l.

t 3 . As an imme(liate d e d u c t i o n from Theorems 3 an(l 5 we m a y state

Theorem 6. I/ [(z) is meromorphic in [ z ] - : 1 tlnd t]~e set I ( / ) is dense on an urc

<x o/ the circumference ]z I l, tl~en S (/)f] I (/) is residual on :r Corollary 6.1. I / m (F (/)N :r then ,S' (/) i.s residual on c(.

For, by Plcssncr's T h e o r e m A, I ( / ) is (lense on ~ if m ( F ( / ) N ~ ) ~0.

Corollary 6.2. I/ I (/) i.~ deJ~se on. ~, then F (]) is o/ category I on ~.

F o r ,~ (/) is residual on :r aIl(l ,~'(/)_~ C F (/) st) t h a t C F (/) is residual ou ~.

This is also a corollary of T h e o r e m 3, since I (/)% C F (]).

~t~. T h e o r e m 5 shows the sets ,~r and I(]) to be topologically equivalent.

T h e y are not, however, metrically equivalent, as can be shown b y k n o w n examples.

F o r instance, in the case of the function ~o ( z ) o f Lusin an(l Privaloff referred to in w 8 above, co (z) takes the radial limit zero p.p., from which it follows t h a t C ~ ' ( w ) is p.p. B u t m F ( w ) : = 0 , for otherwise we should have C~ (~, e~~ in a set of points e ~~ of positive measure which, by Plessner's T h e o r e m B, would i m p l y co (z)---0. There- fore I ( w ) is p.p. an(1 hence, b y T h e o r e m 6, S(~o) is residual an(l, b y T h e o r e m 5, I (w) is also residual. T h u s in this example m I (co) = 2 ~ and m S (o)) - 0 while I (o~) a n d S (w) are b o t h residual sets.

C L U S T E R S E T S O F F U S I C T I O N S M E R O M O R P H I C I N T H E U~TIT C I R C L E 179

VI. Relations between the sets F ( f ) , I ( f ) and S ( f )

t 5 . A s c o u n t e r p a r t to T h e o r e m 6 we have

Theorem 7. I / / ( z ) is meromorphic in I z l < 1 and the set S (/) is dense on an arc

~. o/ the circumlerence [ z ! = 1, then S (/) is re,sidual on ~ and consequently S (/)f~ I (/) is also residual on ~.

For, if S ( / ) is n o t residual on ~ then C S ( / ) 0 ~ is of c a t e g o r y I I a n d i t follows from L e m m a s 1 a n d 2 t h a t there is a F a t o u arc / 3 ~ u . Since, h o w e v e r , / 3 _ ~ C S ( / ) it follows from this t h a t S ( / ) it n o t dense a n d we have a contradiction. T h a t S ( / ) r ) I ([) is residual folh)ws from T h e o r e m 5.

This a r g u m e n t also proves t h a t i/ an arc ~ is contained in W (/), then S (/) 0 I (/) is residual on ~. F o r / 3 _ ~ C W ( / ) so t h a t C W ( / ) 0 ~ is n o t e m p t y if C S ( / ) 0 u is of category I I .

Corollary 7. I[ S (f) is dense on ~, then F (/) is o/ category I on :r

F o r I (/) is (lense and the conclusion follows i m m e d i a t e l y from Corollary 6.2.

I n the same order of ideas we prove

Theorem 8. I / / (z) is meromorphic in [ z l < l and i/ the set I ( / ) (or the set S (/)) is o/ category I on an arc ~r then F (/) is m~trically dense on ~.

For, C I ( / ) 0 :r is residual and hence, b y L e m m a 3, every arc ~,_c~ contains a F~tou arc /3'; and m (E (/) N /3') :- 0. Hence m (F (]) 0 :r ~ m ( E ( / ) A / f f ' ) > 0 , and since :r is an a r b i t r a r y arc in u the theorem is proved.

A n o t h e r metrical relation m a y be noted. If m (S (/)N ~):-: m:r it follows, since S ( [ ) _ ~ C F ( f ) , t h a t m ( F ( / ) N ~ ) = 0 and hence t h a t m ( I ( [ ) f ~ a c ) : = m c t a n d I ( t ) , a n d also therefore S([), is residual on ~. On the other hand, as we saw in the example of the function ~o (z), the condition m (I (/)N 0r m~(, while it implies t h a t I (/) and S (/) are residual, on ~ does n o t imply t h a t m (S (/) 0 :r > 0. I n general terms we m a y say t h a t although S (/) and I (/) are topologically e q u i v a l e n t sets S ( / ) m a y be smaller t h a n I (/) b y a set of measure m I ()'). There is no simple relation, such as inclusion, between the sets I (/) a n d S ( ] ) a n d no simple metrical relation between them. B u t Theorems 3 - 8 and their corollaries have led to a n u m b e r of relations involving some metrical element which m a y be t a b u l a t e d as follows:

For a n y /unction [ (z) meromorphic in [ z ] < 1 and a n y arc ~ o/ the circum/erence

I 1=1,

180 E. F. COLLINGWOOD m ( F (1) n ~) = 0

m ( I ( / ) n ~ ) = m

[ F ( / ) N :r is of category I

~ ' [ S (/) N :r is residual (/) N 0r is residual

(6 a)

m ( S (/) n ~) = m ~ :~ m ( I (/) n ~) = m ~ (6 b ) I (/)N :r of c a t e g o r y I /

-- ! [ ~ F (f) is metrically dense on ~.

!

S (/)N :r of c a t e g o r y I

(6 c)

VII. A t h e o r e m on the m o d u l a r f u n c t i o n

i 6 . Since a set of c a t e g o r y I I is n o t e m p t y Theorems 6 a n d 7 are existence theorems for the sets S ( / ) and I ( / ) u n d e r a p p r o p r i a t e conditions on the set F ( / ) . A particular case of interest is t h a t of the m o d u l a r function /~ (z) defined in the unit circle I z ] < 1. I t is k n o w n t h a t t h e enumerable set of vertices of the m o d u l a r figure on thc circumference Iz[ -- 1 are all F a t o u points1; and it is easily shown t h a t every other point of the circumference is s p a n n e d 1)y an infinity of copies of all three sides of the f u n d a m e n t a l triangle so t h a t , for these points c ~~ C,,(/t, e i~ is non-degenerate and hence ei~ C F(/~). ] t follows t h a t F ( / t ) is cnumcrable, so t h a t I(/~) is p i ) . by Plessner's T h e o r e m A, and hence, b y Theorems 5 and 6, b o t h I(/~) and N(/~ ) are residual on the circumference I z l = 1. We have t h u s proved

Theorem 9. For the modular /unction / ~ (z) regulaz in I zl < I the set S (#) o] points z - e ~~ at which the radial cluster set Cq(tt, e ~~ is total is a residual set on I z l = l ; and the s(t I ( # ) o/ Plessner points, which is p.p. on ] z I = l , is also residual.

R e c u r r i n g to a previous r e m a r k (w 12 above), the t h e o r e m is also t r u e for the sets S~(~)(~t), - ~ 2 "< qJ < 2 ' and 7e Sago) (tt) under suitable restrictions on the curvc ). (0). 2

I t is k n o w n t h a t F ( t t ) is densc a a n d I ( t t ) is p.p. ou [ z [ = l . We now see t h a t S(tt), being residual, is also dense on ]z[ = 1.

1 See, for example, CARATHEODORY [4], p. 275.

2 For example, if ). has an end point on and is non-tangential to ] z l = 1 while I zl is strictly increasing on )..

a C C , p . 140.

C L U S T E R S E T S O F F U N C T I O N S M E R O M O R P H I C I N T H E U N I T C I R C L E 181 Other special cases for which S (/), a n d therefore also I (/), are residual sets are those functions for which F (/) is e m p t y , such as the u n b o u n d e d regular functions which are b o u n d e d on a spiral or on a sequence of closed curves enclosing the origin a n d converging to t h e circumference [z[ = 13

V I I I . B e h a v i o u r i n t h e s e t o f P l e s s n e r p o i n t s

i 7 . We consider t h e chordal cluster sets CQ(r e t~ of /(z) at a given point e ~~ I t has recently been p r o v e d b y Meier z t h a t the set S (/, 0) of values of

2~ 7~

for which CQ(r e t~ is total is of c a t e g o r y I I in the open interval - ~ < ~ < ~ - Our m e t h o d is i m m e d i a t e l y applicable to this problem a n d gives t h e stronger result t h a t S (/, 0) is in fact a residual set in t h e interval. We prove

T h e o r e m 10. I/ /(z) is meromorphic in ] z ] < 1 and et~ (/), then the set S(/, O) o/ values o/ q) /or which C~(~) (/, e t~ is total is a residual set in - 2 < ~ v < 2 .

If, for a given value of ~, Ce(~) (/, e I~ is s u b - t o t a l we can find a circle contained in CCe(v)(/, et~ We again project the w-plane o n t o the w-sphere and proceed ex- a c t l y as in the proof of L e m m a 2. Using the same sequence of t r i a n g u l a r lattices 11, 12, ... ln, ..., n - + ~ , as in t h a t a r g u m e n t ]'n (q~) now denotes the set of values of in - 2 < ~ < 2 such t h a t n is the smallest n u m b e r for which :t ~t CCe(~([, e ~~ con- tains the whole of at least one of the triangles d .... 1 ~< J,~ m (n). We assign to each triangle d .... l _ < v _ < m ( n ) those values of ~vEF=(q) for which dn.,=_CCc(~)(f, et~

Again, all b u t one of these sub-sets of Fn (~), which we denote b y D ~ . , ( q ) , m a y be e m p t y , b u t at least one of t h e m m u s t be n o n - e m p t y a n d a n y two of t h e m m a y h a v e c o m m o n points. We t h u s obtain t h e decomposition

c s ( t , o ) = o U Dn.,(~). (7)

n l < v < m ( n )

If one of the sets On,~ (~v), say Oj, k (~), is of c a t e g o r y I I it follows b y an argu- m e n t similar to the proof of L e m m a 1, which it is n o t necessary to repeat, t h a t there is an angle ~ c o n t a i n e d in - ~ , such t h a t ds.k =-- C C~ ([, e ~~ so t h a t C~ ([, e to) i LuslN and PRIVALOFF [8], pp. 147-150; BAGEMIHL, ERD6S and SEIDEL [1], p. 144; and VA- LIRON [11] and [12], pp. 430-432. Another example is given by BIEBERBACH, 1.C., pp. 152-155.

2 MEIER [9], p. 241.

1 N 2 E . F . C O L L I N G W O O D

is sub-total and consequently e i ~ I ( / ) . Therefore, if e i ~ (/) e v e r y set ~ .... (~) m u s t be of category I in - 2 , ) , a n d the theorem follows from (7).

18. F o r a n y given value of ~, T h e o r e m 10 does n o t of course establish the existence of a n y p o i n t z = e ~~ at which t h e corresponding chordal cluster set C0(~)([, e ~~

of [(z) is total. B u t we have such an existence t h e o r e m in T h e o r e m 6, which, as we have already pointed out, m a y also be p r o v e d for a n y of the sets So(~)(/) and certain of the sets S~co)(/). This leads at once to the following generalisation of T h e o r e m 6. L e t ~t, q)2, "" q~, " " , be a n y enumerable set of values in the interval

21: ~

<:cp~< al~<l d e n o t e b y Sx,,(~) (/) t h e set CI S e ( ~ (/) so t h a t S~,,(~> (/) is t h e set of points z = e ~~ at each of which all t h e chordal cluster sets C,,<+n>(/,e i~ are total. We then have

Theorem l l . I / [(z) is meromorphic in I z l . : 1 a n d i/ either I (/) or a set Sq<~)(/)

~ .

/~)r some q~ in the open interval - 2 : q)':: 2 as dense on an arc :r o / t h e circum/erenee I zl -- l, then, given any. en~Lmerable .set q~, T2, ".. q)n, ... , where - "~2 < ~~ "~'~ ~2' n - : l, 2, ..., the intersection

'~"-~,~.; (I) n I (1) (s)

is residual o~ :~.

F o r b y Theorems 6 and 7, b o t h of which are valid with S~,(~)(/) ill place ~)f

~ (/), I (/) is residual on ~. Hence, t)y T h e o r e m 6, ;~'(,~q,~)(/) is residual on ~ for every valuc of n an(l it follows t h a t the c o m p l e m e n t ([.J CS(,(q,,)(/))U C I ( f ) of (8) bciqg

n

the union of an e n u m c r a b l c set of sets of c a t e g o r y I on ~, is of c a t e g o r y I on :r This proves the theorem.

Some years ago it was p r o v e d b y Kierst a n d Szpilrajn 1 for regular functions t h a t in general, i.e. in a residual sub-space el the space o/ /unctions regular in I zl < 1, the cluster set Co(c) (/, e ~~ is total /or all values el 0 a n d q). More generally, these a u t h o r s p r o v e d t h a t for a class of curves ;t (0) the cluster set C~(o)(/) is total for all values of 0 a n d in a residual sub-space of the space of regular functions. 2 W h a t T h e o r e m 6

J KIERST a n d SZI'ILRAJN [7], p. 2 9 l .

A l t h o u g h KIERST a n d SZPILIIAJN o n l y d i s c u s s regular f u n c t i o n s in t h e circle [ z ~ 1 it is clear t h a t their m e t h o d s will in fact p r o v e t h a t in t h e space of f u n c t i o n s m e r o m o r p h i c in z [ ( 1 the set of funutions ] (z) for which Cg (I) is total for all c u r v e s ). on which [ z [--> I is residual.

CLUSTER SETS OF FU:NCTIONS I~IEROMORPtIIC IN THE UNIT CIRCLE 183 and its generalisations, of which Theorems l 0 an(1 l l are examples, n o w show is t h a t assertions in the same direction, a l t h o u g h less sweeping, can be made u n d e r specified conditions applicable to individual m e r o m o r p h i c fimctions.

I X . A t h e o r e m o n i s o l a t e d e s s e n t i a l s i n g u l a r i t i e s

19. Tile m e t h o d of w 17 is equally applicable to tile ease of a u n i f o r m f u n c t i o n g (z) h a v i n g an isolated essential singularity, which we m a y assume to be at infinity.

We denote b y Ce(~)(g) the cluster set of g(z) as z - > ~ along the r a y ~(~) defined b y z = r e ~'p, 0 ~ < r < oo, and b y C~ (g) the a n g u l a r cluster set of g ( z ) a s z - ~ o o b y sequences {z,~} contained in A. W i t h these definitions we prove

Theorem 12. Suppose that g (z) is non-rational and meromorphic ~or K < [z[ < ~ . The necessury a~zd su//icient co~dition that, given an angle ,4 deft'ned by z ~ - r e *~', q;~ < q~ < q'2, K < r < oo, the angulr duster set C,j (g) o/ g (z) i~ any /J c A shall be total is that the set o/ values o/ ~t ~ in qh < ~ : q~', /or which the radial cblster set C'~,(~, (g) is sub.total is o/ category I.

T h a t the condition is necessary follows at once hy an at)l)lication to the sets (:~,(~,) (g) a n d S (g), the set of values of q, for which C~,(~)(g) is total, of the a r g u m e n t ()f w 17, l)ractically w i t h o u t modification. The sequence of triangular lattices {1,~} is constructe(l an(1 the corresponding subsets 1',~ (g,) of C S (g) are (h,finc(l as in w 17 above and ~vc obtain the (leeomposition

C S (g):: U U ~ ... (~), (',))

1l 1 r : : m u I )

where ~3 ... (~) is now the subset of I'~ (q,) for which C(?t,(v, ) (g) contains the triangle d ... l "_; ~,~ m (**). If a set ~i. k (~) is of c a t e g o r y I I in (g'l, q",) then there is all angle A contained in ~l < ~ ~ ~2 such t h a t dj. k c CC~j (g). T h e angular cluster set U j (g) is thus sub-total a n d the necessity of the con(lition follows. F o r if every (,',1 (g) is total then all the ]0,,~(q,) m u s t be of c a t e g o r y I a n d it follows from (9) t h a t C $ (g) m u s t be of c a t e g o r y I in (~1, q~2).

The con(lition is also sufficient. F o r if it is satisfied every angle A c A contains a r a y z = re*", 0 <_ r < oo, on which ('t,(~)(g) is total; and Ca (g), which contains Ue(r is therefore total.

The theorem can e v i d e n t l y be generalised for d o m a i n s swept o u t b y a curve a (q~) r o t a t e d a b o u t the origin, A ( ~ ) being subject to suitable restrictions, as for example t h a t [z] is strictly increasing on A

(~).

184 :E. F. COLLINGWOOD

Postscript

I t was n o t u n t i l a f t e r t h i s p a p e r was finished t h a t I b e c a m e a w a r e of t w o v e r y r e c e n t p a p e r s of B a g e m i h l a n d S e i d e l [2] a n d [3]. T h e a u t h o r s t h e r e s t u d y p r o b l e m s closely r e l a t e d to t h o s e s t u d i e d here a n d in [2] t h e y p r o v e some of t h e s a m e results.

T h e o r e m 7 (b) a n d C o r o l l a r y 1 of [2] a r e r e s p e c t i v e l y T h e o r e m 1 a n d C o r o l l a r y I of t h e p r e s e n t p a p e r , while T h e o r e m s 7 (a) a n d 7 (b) of [2] are s p e c i a l cases of o u r T h e o r e m 6, a n d T h e o r e m 9 of [2] c o n t a i n s o u r T h e o r e m 10. H o w e v e r , t h e m e t h o d s of these two q u i t e i n d e p e n d e n t i n v e s t i g a t i o n s a r e s u f f i c i e n t l y d i f f e r e n t t o g i v e each of t h e m an i n d e p e n d e n t i n t e r e s t a n d t h e r e a r e a c o n s i d e r a b l e n u m b e r of r e s u l t s w h i c h a r e n o t c o m m o n to b o t h . I n p a r t i c u l a r , o u r T h e o r e m 5 a n d i t s c o n s e q u e n c e s m a y b e m e n t i o n e d . I t h a s t h e r e f o r e s e e m e d b e s t , in s p i t e of t h e o v e r l a p p i n g of t h e re- sults r e f e r r e d t o 1, t o l e a v e t h i s p a p e r u n a l t e r e d e x c e p t for t h e a d d i t i o n of t h i s b r i e f n o t e .

Lilburn Tower, Alnwick, England

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