THEIR EXTENSIONS

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DEFORMATION CLASSES OF MEROMORPHIC FUNCTIONS AND THEIR EXTENSIONS TO INTERIOR TRANSFORMATIONS.

BY

MARSTON MORSE and MAURICE HEINS I n s t i t u t e for Advanced B r o w n U n i v e r s i t y Study PRINCETON, N . J . PROVIDENCE, R. I.

w I. Introduction. The first objective of this paper is to use topological methods and concepts to enlarge the store of knowledge of meromorphic functions. Deformation classes of meromorphic functions are defined. The extension to interior transformations results in new homotopy theorems and contrasts between interior and conformal maps. Earlier papers have shown that there is a con- siderable body of theorems which can be formulated so as to retain meaning and validity after arbitrary homeomorphisms of the z- or w-spheres. Many theorems involving the relations between zeros, poles, and branch point antecedents and the images under f of boundaries are of this character. See Morse and tteins, (I) and Morse (2).

The second objective is to distinguish between the properties of meromorphic functions which are shared by interior transformations and those which are not.

For the transformations from {]z] < I} to the w-sphere which are considered we find no difference 1 between meromorphic functions and interior transformations with respect to the invariants necessary to characterize a deformation class of functions with prescribed zeros, poles, and branch point antecedents. However, sequences [fk(z)] of meromorphie functions properly taken from different deformation classes cover the w-plane in a manner suggestive of the Picard theorem on essential singularities but with no counterpart for sequences of interior transformations. The discovery of such properties points to the problem of finding the non-topological assumptions which must be imposed upon interior transformations in order that they may share the non-topological properties discovered.

' For domains for z o t h e r t h a n t h e d i s c d i f f e r e n c e s m a y arise.

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52 Marston Morse and Maurice Heins.

Topologically we are concerned with the homotopy classes of open simply- connected Riemann surfaces with prescribed properties, conformally with the existence of deformations of prescribed type t h r o u g h meromorphic functions when it is known t h a t such deformations are possible t h r o u g h interior transformations.

We seek to construct models of all deformation classes by ,composing, homeomorphisms of { I z I < I } with an interior transformation with the prescribed zeros, poles, and branch point antecedents. Conformally this is possible only in trivial cases.

Interior transformations are used essentially in the sense of Stoilow. (4) W i t h Whyburn (3) they are ,>interior~ and ~light,. W h y b u r n has studied the underlying point set characteristics of these transformations in a very general setting. The uniformization theorems of Stoilow are found useful. See also, v. Ker6kj~rt6, (5) PP. I 7 3 - - I 8 4 .

The following section gives a summary of the principal results without details or proofs. A detailed exposition follows.

w 2. The problem and the principal re~lts. I n the sense in which we shall use the term an interior transformation w = f ( z ) will be a sense-preserving continuous map of the open disc S{[z I < t} into the complex w-sphere with the following characteristic property. I f Zo is any point on S there exists a sense- preserving homeomorphism 9(z) from a neighborhood N of z 0 to another neighborhood 3~ of Zo with zo fixed, such t h a t the function f [ 9 ( z ) ] = F ( z ) i s analytic on A r except at most for a pole at zo, and is not identically constant. The trans, formation f is said to have a zero or pole at zo if F has a zero or pole at z0- more precisely f is said to have a zero or pole of the order of the zero or pole

of F at z 0.

I f Zo is the antecedent of a branch point of the r-th order of the inverse of F, zo is said to be an antecedent of a branch point of the r-th order of the inverse of f . In any case it is seen t h a t r + I is the number of times a neighborhood of w o ~-f(zo) is covered by f(z) for ] z - zo] sufficiently small.

We shall restrict ourselves to the case in which f has a finite set of zeros

(2. i) So,

. . . a ,

(r > o),

and poles

( 2 .

2)

a r + l , 9 9 a n ( n > I ) 1,

t I n a l l c a s e s m = n + i s h a l l d e n o t e t h e t o t a l n u m b e r of z e r o s a n d p o l e s . T h e c a s e s m = ! a n d m = o a r e t r e a t e d i n w 14.

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Deformation Classes of Meromorphic Functions. 53 with b r a n c h p o i n t a n t e c e d e n t s

(2. s) b~ . . . . , b~, (~ _>-- o),

w i t h n > I u n t i l w 14 is reached. I n the principal s t u d y we shall assume t h a t t h e zeros, poles, a n d branch points are

simple,

t h a t is, have t h e order I. The zeros, poles, a n d branch point a n t e c e d e n t s will form a set of points

( a ) - (ao, 9 9 an, b~ . . . . , bA

t e r m e d

characteristic.

T w o points in the set (a) b o t h of which are zeros, or poles, or b r a n c h p o i n t a n t e c e d e n t s are said to be of

like character.

A n y r e o r d e r i n g of t h e points of (a) in which points of like character are reordered a m o n g them- selves will be t e r m e d

admissible.

E x c e p t in the case g = o, m ~ - 2 we shall use no reordering in which a o is c h a n g e d in relative order.

Admissible fldeformations.

W e shall a d m i t d e f o r m a t i o n s D of f of the form w = F ( z , t ) (lel < i) ( o - _ _ t ~ I)

in which t is the d e f o r m a t i o n p a r a m e t e r a n d

r (z, o) ~ f(z). (I ~ I < i)

W e require t h a t F m a p (z,t) c o n t i n u o u s l y into the w-sphere a n d reduce to a n i n t e r i o r t r a n s f o r m a t i o n for each fixed t. L e t

(a t)

be the characteristic set of F a t t h e time t. W e require t h a t t h e points of (at) vary c o n t i n u o u s l y with t, r e m a i n simple, distinct a n d c o n s t a n t in n u m b e r a n d c h a r a c t e r as t varies f r o m o to I, r e t u r n respectively, w h e n t----I, to some one b u t n o t necessarily the same one of t h e characteristie points of f ( z ) of like character. The t e r m i n a l t r a n s f o r m a t i o n F ( e , I) has the same characteristic set as f w i t h a possible reordering.

I f (a t) is i n d e p e n d e n t of t the d e f o r m a t i o n is t e r m e d

restricted.

I f (a ~ .----(a'), t h e d e f o r m a t i o n is t e r m e d

terminally restricted.

I f (a ~) is an admissible reordering of (a~ D is t e r m e d

semi-restricted. W e

let X denote a n y one of these three types of admissible deformations. I n t e r i o r t r a n s f o r m a t i o n s which a d m i t a d e f o r m a t i o n into each o t h e r of type X will be said to belong to the

same X-deformation class

a n d to be

X-equivalent.

The invariants J(. W e

shall define a set (J) of n n u m b e r s J~ (f, a) (i =-: I . . . . , n) associated respectively with the respective pairs (ao, a~). T h e J,-'s are i n v a r i a n t u n d e r a n y restricted d e f o r m a t i o n of f . The set (J) is t h u s associated with an ordered set (a) in which the first zero a o plays a special role. I f am is a trans-

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f o r m a t i o n w i t h t h e ordered set (a), t h e n all sets (J) belonging to t r a n s f o r m a t i o n s f w i t h t h e same ordered set (a) have the form

( 2 . 4 )

J,(f,

, ) = J , ( F , + r, (i = I . . . . ,

.)

where r~ is an a r b i t r a r y integer. All such sets are realizable.

Topological models with prescribed sets (a) and (J).

One begins by exhibiting an interior t r a n s f o r m a t i o n F with the given ordered set (a). T h e n (2.4) defines t h e ensemble of sets (J) which are

~,associated>>

with (a). A n y one of these sets (J) associated w i t h (a) belongs to a t r a n s f o r m a t i o n f obtainable f r o m F in a simple way.

To o b t a i n these models use is m a d e of sense-preserving h o m e o m o r p h i s m s ~(z) of the disc S = {Izl < I} onto itself-. W e t e r m ~

restricted

if (a) is 'point-wise i n v a r i a n t u n d e r 7,

semi-restricted

if points of (a) are t r a n s f o r m e d into points of (a) of like character. Set n + l---re. Except ~ w h e n m ~ 2 a n d / ~ - o , one can prescribe the ordered set (a) a n d a n y one of the associated sets (J) a n d affirm t h e existence of a semi-restricted h o m e o m o r p h i s m V, d e p e n d e n t on (a) a n d (J), such t h a t t h e f u n c t i o n ~ F ~ has (a) as an ordered characteristic set a n d (J) as its set of invariants. The case m - ~ 2, /~ ~ o is exceptional.

Meromo~Thic r~odels.

The preceding models f are n o t in general meromorphic.

Nevertheless t h e r e exists an explicit f o r m u l a for a m e r o m o r p h i c f u n c t i o n f w i t h a prescribed ordered characteristic set (a) a n d associated i n v a r i a n t s (J).

Restricted deformation classes

C. A necessary a n d sufficient condition t h a t two t r a n s f o r m a t i o n s f l a n d f~ w~th t h e same ordered set (a) be in the same class C is t h a t f~ a n d f~ possess the same i n v a r i a n t s (J) w i t h respect to (a). I f f~ a n d f~, are meromorphic, t h e restricted d e f o r m a t i o n of f~ into f~ which is affirmed to exist can be made t h r o u g h m e r o m o r p h i c functions. This is equally t r u e of the d e f o r m a t i o n classes

C'

a n d

C"

to which we now refer.

Terminally restricted deformation classes

C'. A new i n v a r i a n t is needed here.

Two t r a n s f o r m a t i o n s f w i t h the same ordered set (a) will be said to belong to the same

C'

if a n d only if t h e y have t h e same i n v a r i a n t J1 w i t h r e s p e c t to (~).

Semi-restricted deformation classes C".

I f m > 2 or # > o t h e r e is b u t one class

C"

w i t h t h e given set (a). T h e case / z - - o a n d rn = 2 is a g a i n e x c e p t i o n a l .

Other exceptional cases.

T h e cases in w h i c h m < 2, are excluded u n t i l w 14.

W e have r e q u i r e d t h a t r > o in (2. I), t h u s e x c l u d i n g the possibility t h a t f h a v e poles b u t no zeros. This e x c e p t i o n a l case is r e a d i l y b r o u g h t u n d e r the p r e c e d i n g by r e p l a c i n g f by its reciprocal. T h e case where t h e r e are no poles is a d m i t t e d . Cases in w h i c h t h e c h a r a c t e r i s t i c p o i n t s are n o t simple can be t r e a t e d essentially as in the simple case p r o v i d e d the d e f o r m a t i o n s which are a d m i t t e d are r e q u i r e d to preserve t h e m u l t i p l i c i t y of the respective c h a r a c t e r i s t i c points.

I f one p e r m i t s t h e multiplicities of t h e c h a r a c t e r i s t i c points to c h a n g e by v i r t u e of various t y p e s of coalescence, an i n t e r e s t i n g t h e o r y of d e g e n e r a c y arises a n a l o g o u s to t h e t h e o r y Which describes the d e g e n e r a c y of elliptic f u n c t i o n s i n t o t r i g o n o m e t r i c f u n c t i o n s . I t is p l a n n e d to r e t u r n to this p r o b l e m in a l a t e r paper.

Topological and con formal differences.

A first difference has already been indicated. E x c e p t w h e n m = 2 a n d /~ = o models of all r e s t r i c t e d d e f o r m a t i o n classes with a g i v e n set (a) can be o b t a i n e d as f u n c t i o n s T' by c o m p o s i n g a par- t i c u l a r model F 0 with c h a r a c t e r i s t i c set (a) with suitably chosen semi-restricted h o m e o m o r p h i s m s ,/ of S. T h i s is impossible if one operates only w i t h m e r o m o r - p h i c f u n c t i o n s .

A second difference a p p e a r s in t h e e x t e n t to w h i c h t h e w-sphere is covered by an infinite sequence []~] of t r a n s f o r m a t i o n s w i t h a p r e s c r i b e d set (a) and at m o s t one r e p r e s e n t a t i v e f r o m each r e s t r i c t e d d e f o r m a t i o n class. W e t e r m [f~] a

model sequence.

L e t R be a n y c o n n e c t e d r e g i o n of t h e w-sphere which c o n t a i n s w = ' o and w = oo a n d whose closure does n o t cover the w-sphere. I f t h e models [fk] are n o t r e q u i r e d to be m e r o m o r p h i c a m o d e l sequence [fk] can be defined so as to c o v e r no p o i n t s on t h e c o m p l e m e n t of R and with no subsequence con- v e r g i n g

continuously

on S to o or oo. This is impossible if t h e f u n c t i o n s j~: are m e r o m o r p h i c .

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To give a more revealing statement in the meromorphic case let a point Zo of S be termed a

covering point

of [fk] if corresponding to any arbitrary neighborhood N of z 0 the set of images of -h T under [fk] covers the finite w-plane (w = o excepted) infinitely many times. L e t H be any connected neighborhood on S of the points

(a0, a, . . . = (a)

with the points (a) excluded. Any model sequence [•] of meromorphic functions no subsequence of which ,,converges continuously,, to o or to oo on H possesses at least one covering point on H.

This theorem follows readily from a theorem of Julia on normal families ( s e e Montel (6), p. 37) once the appropriate properties of meromorphic models have been derived. Other theorems concerning the covering points z0 of [fk] are obtained.

We begin the detailed study with the development of homotopy properties of locally simple arcs k and introduce invariants

d(k)

designed to make possible a topological definition of the invariants J~.

P a r t I. T h e T o p o l o g i c a l T h e o r y .

w 3. The difference order d(k) of a locally simple arc k.

The object of this section is to attach a number d(k) to a locally simple, sensed arc k with prescribed end points in such a m a n n e r t h a t

d(k)

remains invariant under a class of deformations of k (to be defined) and characterizes k as representative of its deformation class.

Local simplicity.

See Morse and Heins (I) I. The arcs k which are admitted are ,>represented,) by continuous and locally I - - I images

w(t) ~- u(t) + iv(t) (o < t <= to)

of a line interval (o, to) and shail intersect their end points w(o) and

w(to)

only when t = o and t o respectively. The condition of loeal simplicity implies t h a t there exists a constant e > o sueh t h a t any subarc of k whose diameter is less than e is simple; such a constant e is called a norm

of local simplicity of k.

Any set of loeally simple curves which admit the same norm of local simplicity will be termed

uniformly

locally simple.

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Deformation Classes of Meromorphic Functions. 57 Strictly speaking, w(t) is a representation of k and not identical with k. We admit any other representation of k obtained by mapping the interval (o < t =---to) homeomorphically onto another intervM (o, tl). The arc k can be identified with a class of such representations. The property of local simplicity of an arc k is independent of the representation of k which is used, because the existence of a norm of local simplicity is independent of the representation used.

L e t the end points of k be w = a and w = b respectively.

Admissible deformations of k. We shall admit deformations D of k with the following properties. The arcs of D shall be represented in a locally I - - x way in the form

(3. I) w --- H(t, ~) (o ~ t <--_ to) (o <= Z ~ ~o) where H maps the (t, ~) rectangle continuously into the finite w-plane with

B (o, z) = a n ( t 0 , b

The arc k shall be represented by H ( t , o). The arc k is >-'deformed>> at the >>time>>

into the arc k z represented by H ( t , ~). The arcs k z shall be uniformly locally simple and shall intersect a and b only as end points.

I f the requirement of uniform local simplicity were replaced by the requirement t h a t the arcs k x be separately locally simple, there would be but one deformation class of arcs with the end points a and b, instead of the countably infinite set of deformation classes which actually exists. 1

Any two arcs k ~ appearing in the above deformation are said to be in the same deformation class. Actually the deformation is one of >)representations*, but it is immediately obvious t h a t any two representations of the same arc can be admissibly deformed into each other. See (i) I p. 6o3. Accordingly the proper~y of two arcs of being in the same deformation class is independent of their representations.

To avoid misunderstanding, the following should be pointed out. The possibility t h a t a ~--b is admitted. Let ~a and k~ be proper simple subarcs of k with a the initial point of ka and b the terminal point of kb. The arcs ka and kb can intersect in infinitely many points, even coincide. Fig. I shows four examples of arcs k in the same deformation class in the case a = b. The figures must of course be superimposed so t h a t the point w = a coincides with w ~ b in all four cases. The lower right curve is designed to indicate the possibility of a curve

i Cf. L e m m a 3 . 3 .

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w ~ Ot

Marston Morse and Mauriee Heins.

58

F i g u r e I.

W ~ O .

in the given deformation class with a simple spiral terminal arc. A n o t h e r figure could be drawn indicating an arc in t h e same deformation class with spiral like simple subarcs at both ends. Reversed in sense the f o u r arcs in Fig. I belong to a second and different deformation class.

The regular case a ~b.

The arc k is termed

regular

if it admits a representation

w(t)

in which

w'(t)

exists, is continuous and is n e v e r zero. F o r m a n y purposes it will be convenient to measure angles in

rotational units,

t h a t is, in units which equal 2 z radians. The algebraic i n c r e m e n t in

2)

w'(t)

(o _-<

t _-< to)

2:7V

as t increases from o to t o and arg

w'(t)

varies continuously will be denoted by P(k).

The angle 19 is in r o t a t i o n a l units and represents the total angular variation of the t a n g e n t to k as the point of tangency

w(t)

traverses k. L e t e be either end point of k. T h e limiting algebraic increment of

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Deformation Claases of Meromorphic Functions. 59 arg [w (t) ^-^- c] [O < t < to]

2 ~

as t increases from o to to and the a r g u m e n t varies continuously will be d e n o t e d by Q~(k).

I n the regular ease with a ~ b we tentatively define d(k) by the equation

(3.3) d(k) = P(k) - - Qa(k) - - Qb(k) (a ~ b)

and show that d(k) is an integer.

The n u m b e r d(k) equals the r o t a t i o n a l units in the t o t a l a n g u l a r variation of a u n i t vector X which varies continuously as follows from

b - - a

(3.4) ] b - - a ]

back to the same vector. L e t X start with the vector (3.4) and coincide with

,w(t) - a (to > t > o)

(3.5) I w ( t ) - al

as t decreases from t o to o, excluding o'. L e t X then coincide with w'(t)

(3.6) [ w'(t) 1 (o _-< t =< to)

as t increases f r o m o to to. Finally let X coincide with

b - w (t) (to > t > o)

(3.7) I ~ - w (t)~ =

as t decreases f r o m to to o. The initial and final vectors coincide with (3.4).

The algebraic i n c r e m e n t s of a r g ( ~ ) c o r r e s p o n d i n g to the variations associated with (3.5), (3.6) and (3.7) respectively are

- q o , •, - - Q b ,

so t h a t the r e s u l t a n t algebraic i n c r e m e n t in angle is given by (3-3). The variation in X is c o n t i n u o u s so t h a t the s t a t e m e n t in italics follows.

The general case a ~ b. The arc k is no longer a s s u m e d regular. A subarc of k will be defined by an interval (a, $) for t. I f this subarc is simple the vector

(3.8) w (~) - - w (a)

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60 Marston Morse and Maurice Heins.

has a well defined direction. This direction will vary continuously with a and v so long as the arc (a, ~) remains simple and a < ~. W e term such a variation an admissible chord variation. In such a variation we suppose that the angle

(3.9) I _ arg [w (~) -- w (a)]

2 ~

has been chosen so as to vary continuously with a and ~. For an admissible chord variation the algebraic increment of (3.9) depends only on the initial and finaI simple subarcs.

Let ka and kb be respectively proper simple subarcs of k of which the initial point of ka is a and the terminal point of kb is b. Let the chord subtending ka vary admissibly into the chord subtending hb. Let

P (k, ka, k~)

represent the resulting algebraic increment of (3.9). Let the algebraic increment of

(3. IO) I arg [w(t)-- a]

2 ~

as t increases from its terminal value ta on ~a to to be denoted by Qa(k, ka) and let the algebraic increment of

(3. ii) I__ arg [w(t) - b]

2 ~

as t increases from o to its initial value tb on hb be denoted by Qb(k, hb).

The difference order d(h) when a # b is defined by the equation

(3. i2) = P(h, ho, (a b)

W e state the lemma.

L e m m a 3.1. The difference order d(h) as defined by (3.12), where a # b, is an integer which is independent of the choice of ha and kb among proper simple subarcs of k with end points as prescribed.

The proof t h a t d(h) is an integer is essentially the same as in the regular case. Aa before, d(k) measures the angular variation of a unit vector which initially and terminally coincides with (3.4) and may be broken up into the variatiuns which define --Qa, P, and - - Q b respectively in (3.12).

I t should be noted that any one of these three component angular variations may be arbitrarily large in numerical value and increase without limit as ha or kb tend to a or b respectively. This would certainly happen if h had spiral like terminal simple subarcs. W e have the following lemma.

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L e m m a 3 . 2 .

(3.

consistent with the earlier definition (3.3).

T h e f o l l o w i n g l e m m a is also i m m e d i a t e l y obvious.

Deformation Classes of Meromorphic Functions.

I f k is regular

d (k) -~ P -- Q~ - Qb (a ~ b)

61

L e m m a 3 . 3 . The d~erenee order d(k) (a ~ b) is i~dependent of any admissible deformation of k. There are accordingly at least as many deformation classes as there are different values of d(k) for admissible arcs joini~g a to b.

F o r any i n t e g e r m a ~model,~ arc K ~ j o i n i n g a to b a n d with d ( k ) = m can be defined as follows. F o r n ~ - o t a k e K o as t h e s t r a i g h t line s e g m e n t f r o m a to b.

L e t c be t h e m i d p o i n t of K o a n d C be a u n i t circle t a n g e n t t o K o a t c a n d to t h e l e f t of K o. T o define K~ f o r n > o t r a c e K o f r o m a to c, t h e n t r a c e C n t i m e s in t h e positive sense, a n d finally t r a c e K o f r o m c t o b. T a k e K - n as t h e reflec- t i o n of Kn in K 0. T h a t d(K,n)= m follows f r o m (3. I3). Since d(k) is i n v a r i a n t u n d e r admissible d e f o r m a t i o n s n o t w o of t h e models Km are in t h e same de- f o r m a t i o n class. I t can be s h o w n 1 t h a t each admissible c u r v e k w h i c h joins a to b a n d f o r w h i c h d(k)---m is admissibly d e f o r m a b l e into K ~ .

T h e case b = co. W e suppose t h a t a is finite as previously, a n d m a k e t h e obvious e x t e n s i o n s of p r e v i o u s definitions as follows.

T h e arc k m a y be g i v e n in ~he w-plane. B y c o n v e n t i o n it j o i n s w ~ - a to w = c o if its closure k o n t h e w-sphere joins w ~ a to w~--co. I t is t e r m e d locally simple if k is locally simple on t h e w-sphere. Admissible d e f o r m a t i o n s are defined as previously b u t on t h e w-sphere. I f k is locally simple t h e r e exists a value ~ w i t h o < z < to such t h a t t h e s u b a r c 9 ~ t < t o is simple in t h e finite w-plane. On t h i s arc ]w(t)] becomes infinite as t t e n d s to t o.

T o define d (k) let ld r e p r e s e n t t h e subarc (o, s) of k, w i t h o < s < t 0. T h e n d(/d) is well defined. I t is clearly i n d e p e n d e n t of s p r o v i d e d s > ~. W e ac- c o r d i n g l y set

d (k) ---- d (k') (~ < s < to).

T h a t d(k) is i n v ~ r i a n t u n d e r admissible d e f o r m a t i o n s follows as previously.

T h e arc k is t e r m e d regular if its closure k on t h e w-sphere is r e g u l a r ; one t h e n has t h e lemma.

1 W e m a k e n o use of t h i s t h e o r e m a n d accordingly o m i t t h e proof.

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6~

L e m m a 3 . 4 .

(3.

Marston Morse and Maurice Heins.

I n ca,ve ~ is regular on the w.sphere, a is .finite and b~-oo, then

d(]c) = ~D(]C)_ ~a(]c).

I n case ~ is r e g u l a r on t h e w-sphere k has a definite asymptote in t h e finite w-plane as t t e n d s to to. T h e subare k' of k is r e g u l a r , and if w ( t ) r e p r e s e n t s k,

= Q o ( k - -

On l e t t i n g s t e n d to to t h e last t e r m t e n d s to zero a n d t h e first two t e r m s t e n d to t h e c o r r e s p o n d i n g t e r m s in (3. I4). T h u s L e m m a 3 . 4 holds as stated.

The case a ~---b. W e h e r e suppose t h a t a is finite. T h e values

Qa (k, ]ca) Qb (]C, kb)

are u n d e f i n e d since t h e y involve t h e null v e c t o r b - - a . Moreover, t h e value of d(]c) which would be o b t a i n e d by t a k i n g an a p p r o p r i a t e l i m i t as a t e n d s to b is n o t t h e i n v a r i a n t which is useful. I n s t e a d we proceed as follows.

I n t h e r e g u l a r case one sets

(3-IS) d(k) = P(k) - - Qa(]C). (a = b)

I n t h e g e n e r a l case let

Qa (k, ]ca, ]cb)

e q u a l t h e a l g e b r a i c i n c r e m e n t of

(3 16) I a r g [ w ( t ) - - a]

2 ~

as t varies monotonically f r o m fls t e r m i n a l value t. on ka to its initial v a l u e tb on kb. In the g e n e r a l case one sets

(3. I7) d(]c) = ~ ( k , ka, kb) - Qa(k, ka,]cb). (a = b) T h e first r e l e v a n t facts are as follows:

L e m m a 3 . 5 . The value of d (]C) when a ~ b equals ~- rood z. I t is independent of the choice of ]ca and ]cb among proper simple subarcs of ]C with end points as prescribed, and is invariant u~der admissible deformations of ]C.

T h e a n g u l a r v a r i a t i o n which defines d(k) is t h a t o f a v e c t o r Y which s t a r t s with w ( t ~ ) - - a a n d ends w i t h this v e c t o r r e v e r s e d in sense, i n f a c t it is suffi- c i e n t if Y first varies admissibly as a c h o r d f r o m its initial p o s i t i o n to t h e v e c t o r b ~ w ( t b ) sub~ending ]co; this v a r i a t i o n gives P in (3. I7)- One c o n t i n u e s with a v a r i a t i o n of

b - -

w (t)

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Deformation Classes of Meromorphie Functions. 63 in w h i c h t d e c r e a s e s f r o m tb to ta a n d in w h i c h t h e a n g u l a r v a r i a t i o n is m e a s u r e d by Qa in (3- I7). B u t t h e t e r m i n a l v e c t o r Y coincides with t h e i n i t i a l v e c t o r Y r e v e r s e d in sense so t h a t d(k), m e a s u r e d in r o t a t i o n a l u n i t s , equals 1 m o d I.

T h e i n d e p e n d e n c e of d(lc) of t h e choice of ]Ca a n d kb is clear as is its in- v a r i a n c e u n d e r a d m i s s i b l e d e f o r m a t i o n s .

W e n o t e t h e following.

L e m m a 3 . 6 . I l k is regular a~zd i r a = b (finite), or i r a is finite and b = o o , then

(3. 18) d(]c) = P ( k ) - - Qa(k).

Models f o r t h e d i f f e r e n t d e f o r m a t i o n classes w h e n b = co a r e o b t a i n a b l e f r o m t h o s e f o r which b = b~ (b~ finite) by m a k i n g a d i r e c t l y c o n f o r m a l t r a n s f o r m a t i o n of t h e w-sphere which leaves a fixed a n d c a r r i e s b t to oo.

I n case a = b (finite) t h e values of d(k) are f o u n d to be - - 3 I I 3

2 2 2 2

A positively s e n s e d circle C t h r o u g h a h a s a difference o r d e r 89 R e v e r s i n g sense a l w a y s c h a n g e s t h e sign of t h e difference order. T o o b t a i n a c u r v e K r w i t h a

2 r + l

difference o r d e r ,n . . . . , a n d w i t h r > o, one a t t a c h e s a small p o s i t i v e l y

2

s e n s e d circle C1 to C w i t h i n C, a n d t a n g e n t to C a t s o m e p o i n t c o t h e r t h a n a = b; one t h e n t r a c e s C u n t i l e is r e a c h e d , t h e n t r a c e s C1 r t i m e s in t h e p o s i t i v e sense, a n d c o n t i n u e s to b on C. T o o b t a i n a c u r v e w i t h difference o r d e r - - n o n e c a n r e v e r s e t h e sense of /~%, or m o r e s y m m e t r i c a l l y reflect K~ in t h e t a n g e n t to C a t a .

I t c a n be s h o w n t h a t t h e s e m o d e l s r e p r e s e n t all possible d e f o r m a t i o n classes of a d m i s s i b l e c u r v e s w h e n a = b. W e shall n o t use this fact.

w 4. Three deformation lemmas. L e t h be a s i m p l e arc j o i n i n g t w o d i f f e r e n t p o i n t s z 1 a n d z~ in t h e finite z-plane. ~Admissible~> d e f o r m a t i o n s of such a n arc k e e p z~ a n d z~ fixed. I f in a d d i t i o n only simple arcs are e m p l o y e d , t h e d e f o r m a - tion will be t e r m e d a n isotopic d e f o r m a t i o n of h. A first l e m m a is as follows.

L e m m a 4. 1. Any simple are h, joining zl and z2 (zl ~ z~) i~ the finite z-plane, can be isotopieally deformed on an arbitrary neighborhood N o f h into a simple, regular, m~al?ltie are joining zl to z~.

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64 Marston Morse and Maurice Iteins.

We begin by proving the following:

(a) The arc h can be isotopically deformed on • into a simple arc h* on which sufficiently restricted terminal subarcs are straight.

For simplicity take ~ 1 = o - Choose e so t h a t o < 2 e < ] z ~ l . L e t C be the circle I z l = e and E the open disc {Izl < e } . L e t hi be a maximal connected subarc of h on E with z = o as an initial point. L e t h~ be a simple open arc of which hx is a subarc, which lies on E and whose closure is an arc joining diametrically opposite points of C. There exists a sense-preserving homeomorphism T of /~ which leaves z = o and C pointwise fixed, and maps h2 onto a diameter of /~'. I t follows from a theorem of Tietze (7) t h a t T may be generated by an isotopic deformation z/ of /g from the identity leaving C and z = o pointwise fixed.

I f e is sufficiently small, J will deform h V ~' only on _A r and yield an image of h with a straight initial subarc. Such a deformation of h will be isotopic.

Neighboring z~, h can be similarly deformed, so t h a t (a) follows.

To complete the proof of the lemma, let g be a closed J o r d a n curve of which the arc h* in (a) is a subarc, and which is analytic and regular neighboring zx and z~. Let R be the J o r d a n region bounded by g. Let ~ be mapped homeomorphically onto a circular disc, conformally at points of /~. On the disc the pencil of circles through the images of z~ and z~ will have antecedents on /~ which will suffice to deform h* isotopically on ~V into a simple, regular, analytic arc joining z~ to z~.

This completes the proof of the lemma.

In deriving deformation theorems for h when h joins z~ to z~ on S and zx~z~, no generality is lost if it is supposed t h a t a is real and positive and that, with a < I,

Zl - - , z. = , (a o).

For S can be mapped conformally onto itself so t h a t z I and z~ go into - - a and a respectively for a suitable value of a.

A non-singular ~ran,~ormation of coordinates. L e t the z-plane be referred to polar coordinates (r,O). For the above a and for each constant b on the interval o ~ b < I a transformation from polar coordinates (0, 9) to (r, 6) will be deflated by the equations

(4.!) r = , o _ 0 2 ~ , 8 - - - ~ (o <: Q~be < I).

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Deformation Classes of Meromorphic Functions. 65 F o r b = o , (4. I) r e d u c e s to t h e i d e n t i t y . F o r b ~ o t h e i n t e r v a l o ~ r < c~ a n d t h e i n t e r v a l

(4.2) o=<e<~

I

c o r r e s p o n d in a I - - I m a n n e r . T h e circle Q = a c o r r e s p o n d s to t h e circle r = a.

S u b j e c t to (4-2) t h e r e c t a n g u l a r t r a n s f o r m a t i o n of c o o r d i n a t e s defined by (4. I) is I - - I , a n a l y t i c a n d n o n - s i n g u l a r . T h e circle #--- I c o r r e s p o n d s to t h e circle f o r w h i c h

I - - a 2 b e

(4.3) r = I - b e . .

W e shall m a k e use of t h e t r a n s f o r m a t i o n (4. I) a n d p r o v e t h e f o l l o w i n g l e m m a . L e m m a 4.9.. A simple, regular, analytic are h joining - - a to a on S can be isotopicaIly deformed on S through regular anatglic ares h ~ i~do a circular are j o i M n g - - a to a on S m~der a deformation in which the point with le~gth parameter x on h corresponds to a p o i n t

(4.4) z = z (s, t) [ o~

at the time t, where z (s, t) is m~alytic in the real rariables (s, t), and z, ~ o.

To define t h e d e f o r m a t i o n (4.4) let g be a J o r d a n era're on S which includes h as a s u b a r c a n d w h i c h is a n a l y t i c a n d r e g u l a r n e i g h b o r i n g a a n d - - a . L e t R be t h e J o r d a n r e g i o n on S b o u n d e d by g. G e t C be a circle on _R a n d l e t R, be t h e s u b r e g i o n of R e x t e r i o r to C. T h e r e g i o n //1 can be m a p p e d I -- I a n d c o n f o r m a l l y on an a n n u l u s A in such a m a n n e r t h a t t h e m a p p i n g c a n be ex- t e n d e d I - - I a n d c o n f o r m a l l y o v e r C a n d h. S u p p o s e t h a t t h e i m a g e k of h is on t h e o u t e r c i r c u l a r b o u n d a r y of A. W e shall d e f o r m k t h r o u g h c o n c e n t r i c c i r c u l a r arcs kt, o =< t =< I, on A into a p r e s c r i b e d arc X" 1 on t h e i n n e r c i r c u l a r b o u n d a r y of A. S u c h a d e f o r m a t i o n is r e a d i l y defined in t e r m s of t h e p o l a r c o o r d i n a t e s of t h e a n n u l u s so as to h a v e a r e g u l a r a n a l y t i c r e p r e s e n t a t i o n in t e r m s of t a n d t h e are l e n g t h on k. T h e a n t e c e d e n t ht on R , of la on A will d e f o r m h = ho i n t o a s u b a r c h, of C.

D u r i n g t h e d e f o r m a t i o n ht of h t h e end p o i n t s of ht move. T o r e m e d y t h i s d e f e c t let ht be c a r r i e d by a l i n e a r t r a n s f o r m a t i o n a z + d (unique) i n t o an arc Ht j o i n i n g - - a to a. N o t e t h a t H o = h o = h , a n d t h a t //1 will be on S p r o v i d e d t h e a b o v e arc k, h a s b e e n chosen sufficiently short, a n d this we s u p p o s e done.

5 6 1 4 9 1 1 1 2 Acta mathem~tica. 79

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T h e d e f o r m a t i o n /It satisfies t h e i e m m a e x c e p t t h a t H m a y n o t r e m a i n on S, a l t h o u g h it begins a n d ends on S. W e shall a c c o r d i n g l y m o d i f y /art as follows.

L e t r(t) be t h e m a x i m u m value of r on Ht. N o t e t h a t r(o) a n d r ( I ) are < I.

L e t [t] be t h e set of values of t on which r ( t ) ~ I. L e t b(t) be a real a n a l y t i c f u n c t i o n o f t f o r o ~ t = < I such t h a t

(4.5) o <: b(t) < I

a n d so n e a r I on [t] t h a t

(4.6) r (t) <

t ( o ) = = o ,

I ^-^- a" b ~ (t)

- t, (t) ( 0 ~ t ~ I).

W i t h b(t) so c h o s e n let Tt be t h e t r a n s f o r m a t i o n f r o m t h e z-plane of (r, 8) defined by t h e inverse of (4. I) when b = b ( t ) , and let h t be t h e image T t H t of He at t h e t i m e t. I t follows f r o m t h e choice of b(t) a n d f r o m (4.6) t h a t r I o n h t, so t h a t h t is on S. Moreover, h ~ a n d f o r e v e r y t t h e e n d points of h t a r e a and - - a respectively. T h e arc h ~ is t h e circular arc //1.

T h e d e f o r m a t i o n h t, s u i t a b l y r e p r e s e n t e d , satisfies t h e lemma. T h e represen- t a t i o n of h t is c o m p l e t e l y d e t e r m i n e d by the r e q u i r e m e n t t h a t z(s, t ) r e p r e s e n t t h e p o i n t on h t at t h e t i m e t into which t h e p o i n t s on h has been d e f o r m e d . T h e c o n d i t i o n z~ ~ o is obviously satisfied a n d t h e p r o o f of t h e l e m m a is complete.

(b) _rn the preceding lemma at most a finite number of arcs h t pass through m~y point of S not on h.

To verify (b) let Zo be a p o i n t on S n o t on h. T h e set of pairs (,~,t) on t h e (s, t) r e c t a n g l e w h i c h satisfy t h e c o n d i t i o n z(s, t ) = Z o is e m p t y , or consists of a finite n u m b e r of pairs, or includes a t least one a n a l y t i c arc s = s(t). I n t h e last case t h e f a c t t h a t z,(s, t) ~ o insures t h a t t h e arc s(t) can be c o n t i n u e d analytically, in e i t h e r sense, a n d in p a r t i c u l a r in t h e sense of d e c r e a s i n g t u n t i l a b o u n d a r y p o i n t (s*, to) of t h e (s, t) re.ctangle is reached. B u t z(s*, to) is t h e n a p o i n t of h, so t h a t z0 is a p o i n t of h, c o n t r a r y to hypothesis. H e n c e (b) holds as stated.

T h e f o l l o w i n g l e m m a is a c o n s e q u e n c e of L e m m a 4 . 2 a n d (b).

L e m m a 4. 3. A n y two simple, regular, analytic arcs h I and h e which join - - a and a on S (a ~ o) can be isotopically deformed into each other on S through simple, regular, analytic arcs ~o more than a finite ~umber of which pass through any point of S not on h 1 or h.2.

T h e arc hi, i -= 1,2, c a n be d e f o r m e d on S in the m a n n e r s t a t e d in L e m m a 4. I a n d in (b), i n t o a circular arc k~ j o i n i n g ~ a to a. B u t kl can be d e f o r m e d i n t o k~ t h r o u g h a pencil of circles j o i n i n g - - a to a on S. L e m m a 4-3 follows.

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Deformation Classes of Meromorphic Functions. 67 w 5. The invariants J~. Before c o m i n g to t h e definition of the i n v a r i a n t s Ji a t h e o r e m on interior t r a n s f o r m a t i o n s will be recalled. See ref. (I) I I p . 653. L e t g be a locally simple, sensed, closed plane curve: w -~ w(t), with w(t + 2 z) ~-- w(t).

I f el is a sufficiently small positive constant, a n d o < e < el

I arg [w (t + e) - - w (t)] (o _--<_ t =< 2 ~)

27g

will be well defined, ~ a n d as t increases f r o m o to 2 ~r, will change by an i n t e g e r p(g) i n d e p e n d e n t of e < e r W e t e r m p the angular order of g. I f g does n o t pass t h r o u g h w--~ o, its o r d i n a r y order with respect to w ~ o will be d e n o t e d by q(g).

The t h e o r e m which we shall use is as follows:

T h e o r e m 5.1. Let B be a closed Jordan curve in the z-plane with interior G and let F ( z ) be an interior transformation of some region R which includes G in its interior, and which maps t~ into the w-sphere. I f B is fi'ee from branch point an- tecedents and has an image g which does not intersect w ~ o or oo, then

(5. i) n(o) + n ( o ~ ) - - ~ = ~ + q ( ~ ) - - p ( ~ )

where n (o), n (oo) and ~ are respectively the numbers of zeros, poles, and branch point antecedents of F on G counting these points with their multiplicities.

To come to the definition of the i n v a r i a n t s Ji, let f be an i n t e r i o r trans- f o r m a t i o n f r o m S - - - - { I z [ < I} to the w-sphere wi~h t h e characteristic set

(~) = (a0, ~1, L e t hi be a simple curve j o i n i n g a o to to be of the same topological type if

. . . , a,~, bl . . . . , b,) (m----n + I).

at on S with i > o. Two curves hi will be said t h e y can be isotopically d e f o r m e d into each other on S w i t h o u t i n t e r s e c t i n g the set (a) other t h a n in h / s end points a 0 a n d ai.

L e t h~ denote the image of h~- u n d e r jr. The curve h~ will join f(ao) to f(at) in the w-plane a n d be locally sflnple. I f h~ is isotopically d e f o r m e d for o ~ t ~ I t h r o u g h curves of the same topological type, h~ will be admissibly d e f o r m e d in t h e sense of w 3 t h r o u g h a u n i f o r m l y locally simple family of arcs j o i n i n g f(ao) to f(ai) a n d i n t e r s e c t i n g f(ao) and f(a~) only as end points. The difference order d(h~) is accordingly i n v a r i a n t u n d e r such deformations.

H o w e v e r d(h~) will c h a n g e in general with the topological t y p e of hi. I t is possible to define a n o t h e r f u n c t i o n V(h~) of hi which changes w i t h the type of hi exactly as does d(h{). To t h a t end set

H e r e a n d e l s e w h e r e t h e a r g u m e n t of a n y c o n t i n u o u s l y v a r y i n g n o n - n u l l f u n c t i o n F w i l l b e t a k e n as a b r a n c h w h i c h v a r i e s c o n t i n u o u s l y w i t h _~:

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A(~',ff) = ( g - - ao) ( g - - al) . . . ( 2 ' - - an)

(5. :) B ( ~ , ,~) = (~ - - b~) (.~ - - t . ) . . . (~ - - t , ) (~, > o)

B ( ~ , , ) = ~ (~ = o)

. . . . - A - ( ~ ] - ~ y . . . (i > o).

The r i g h t member of (5.3) h a s a removable s i n g u l a r i t y at z = a o a n d a t z = ai.

W e suppose t h a t Ci(z, a) takes on its l i m i t i n g values at a o and ai so t h a t

(5.4) C} (ao, a) = (a~ - - a~) B (ao, a)

A' (ao, ,~)

--

ao) B (ai, a)

(5.5) C~ (a,, a) --- (ai A ' (ai, a)

(i > o)

Corresponding to a variation of z along hi set

( s . 6 ) V(hi) = [ a r g C~(z, a)l z=ai.

" ~ Z ~ a o

Regardless of the a r g u m e n t s used

(5.7)'

V(hi) ~- ~ [arg Ci(ai, a) -- arg C/(a o, ~)] (mod I).

2~V

W e shall prove t h e following t h e o r e m : T h e o r e m 5.2. The value of the difference

(ft. 7) d(h~) - - V(hi) (i = i, . . . , n)

is independent of hi am'ong simple curves which joi,n a o to ai without intersecting the other points o f the characteristic set (a).

The value of t h e difference (5.7) is clearly i n d e p e n d e n t of isotopic deformations of hi t h r o u g h curves of t h e same topological type, since this is true of b o t h t e r m s in (5.7). By virtue of L e m m a 4. I we can accordingly r e s t r i c t a t t e n t i o n to arcs hi which are admissible in the l e m m a a n d in a d d i t i o n are regular a n d analytic. I f h ~ a n d h~ are two such curves we seek to prove t h a t (5.7) has t h e same value for h~" as for h ~

I n accordance with L e m m a 4.3 there exists a n isotopic d e f o r m a t i o n of hi into h~ t h r o u g h simple, regular, analytic arcs h~ ( o ~ t ~ I) such t h a t h~ intersects t h e set (a)--(a0, ai) f o r at most a finite set of values t o of t. I t is only as t passes t h r o u g h such a value t o t h a t

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D e f o r m a t i o n Classes of M e r o m o r p h i c F u n c t i o n s . 69

J

a. d ai

Figure 2.

(s. s) - r

c o u l d p o s s i b l y c h a n g e . S u p p o s e t h a t h~ ~ p a s s e s t h r o u g h j u s t o n e p o i n t z = c o f t h e s e t ( a ) - - ( a 0 , at). T h e p r o o f in t h e c a s e in w h i c h t h e r e a r e s e v e r a l p o i n t s z ~ - c o n hti o will be s e e n t o be similar. W e s h a l l s h o w t h a t t h e r e is n o n e t c h a n g e in (5.8) as t - - t o c h a n g e s sign.

L e t

h', h",

a n d h ~ be r e s p e c t i v e l y t h e a r c s h~ f o r w h i c h t : t o - e, t o + e a n d t o.

F o r e 1 s u f f i c i e n t l y s m a l l a n d o < e < el, t h e c u r v e s h' a n d h " will i n t e r s e c t (a) in a0 a n d at o n l y . W e s u p p o s e t h a t e < e~. W i t h o u t loss o f g e n e r a l i t y w e c a n s u p p o s e t h a t

h'

p a s s e s ~ z ~ - c t o t h e r i g h t of h ~ I f

h"

l i k e w i s e p a s s e s z : c t o t h e r i g h t of h ~ it is p o s s i b l e t o d e f o r m h' i n t o

h"

w i t h o u t i n t e r s e c t i n g z : c.

(One m o v e s e a c h p o i n t o f h' a l o n g a n o r m a l ~ t o h ~ u n t i l

h"

is met.) I n t h i s case (5.7) h a s t h e s a m e v a l u e o n

h'

as o n h " .

S u p p o s e t h e n t h a t

h"

p a s s e s z : c t o t h e l e f t o f h ~ W i t h o u t c h a n g i n g t h e t o p o l o g i c a l t y p e o f

h'

o n e c a n d e f o r m t h e p o i n t s o f

h'

a l o n g n o r m a l s t o h ~ so t h a t

h'

c o m e s t o c o i n c i d e w i t h h o e x c e p t on a s h o r t o p e n a r c k' w h i c h lies t o t h e r i g h t o f h ~ n e i g h b o r i n g z : c . S i m i l a r l y o n e c a n d e f o r m

h"

so t h a t it c o m e s t o c o i n c i d e w i t h h o e x c e p t o n a s h o r t o p e n a r c ~" w h i c h lies t o t h e l e f t o f h ~ n e i g h b o r i n g z : c. W e c a n a l s o s u p p o s e t h a t t h e e n d p o i n t s o f

k'

a n d

k"

o n

h o

c o i n c i d e i n p o i n t s P a n d Q. See F i g . 2.

L e t M a n d N be p o i n t s o f h ~ s u c h t h a t t h e a r c ( M N ) o f h ~ c o n t a i n s t h e a r c ( P Q ) o f h ~ o n its i n t e r i o r . L e t fl be a s i m p l e o p e n a r c w h i c h j o i n s M t o ZT t o t h e l e f t o f h " so n e a r h " t h a t

B " : fl(MP)k" (Q N)

1 More definitely, we suppose that h' intersects the normal to h ~ at z = c to the right of ho.

2 Provided el is sufficiently ,small.

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is a J o r d a n curve with no points of (a) on the closure of its interior. W e refer to the subarcs ( M P ) and ( Q N ) of h ~ The closed curve

B'--- fl(MP)k' (Q N)

is then simple and contains no point of (a) other than z = c. L e t

B'

and

B"

be taken in their positive senses relative to their interiors. Let

G"

be the region bounded by

B',

and G" the region bounded by

B".

W e shall apply Theorem 5. I

~' G-"

t o f a s defined on G , and t o f a s defined on .

Case I. f o n

G'.

Let

g'be

the image of B ' u n d e r f. The p o i n t z = c i s t h e only zero, pole, or branch point antecedent on G'. The numbers n(o), n ( c ~ ) a n d tt refer to z = c. One only of these numbers differs from zero. In accordance with Theorem 5. I,

(5.9) ,(o) + n ( o o ) - - t t - ~ I +

q(g')--p(g').

Case II. f on G". L e t g" be the image of B ' under f . There are no zeros, poles or branch point antecedents on G" so that

(5. ,o) o = I + q(g")--p(g").

From (5.9) and (5. Io) one obtains the relation

(5. ~ ) . ( o ) +

, ( ~ ) - ~

= [ p ( g " ) - q(g")] - [ p ( g ' ) - q (g')], Equation (5.1 I) will give us our basic equality.

On taking account of the fact that B ' coincides with

B"

except along the arcs k' and

k"

respectively, and that

h'

(as altered) similarly coincides with

h"

(as altered) except along k' and

k"

respectively, it appears that the right member of (5. I I) has the value

(5. ~2) d(h ''J) -- d(h 'f)

in accordance with the definition of the difference order d.

On the other hand,

F'(h')- V(h")

reduces to the variation of arg Ct along the closed curve

k'k",

and so equals the number of zeros minus the number of poles of Ct within this curve. Thus

F

(5. i3) r ( h ' ) - - r ( h " ) = ~ - - . ( o ) - - . ( o o ) in accordance with the ~tefinition of Ct. From (5. I I) then

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Deformation Classes of Meromorphic Functions. 71 (S. I4) d(h ''f) -- d(h 'f) = V(h") -- V (h').

There is thus no change in (5.7) as t passes through to, and the theorem follows.

The invariants Ji. As suggested by Theorem 5 . 2 we set

(5" I5) Ji(f,a)----d(h{)--V(hi) ( i = i . . . . , n)

for any simple arc hi which joins a o to a~ without intersecting ( a ) - (ao, a~). The numbers J~ are independent of the ehoice of hi among admissible arcs hi and of restricted deformations o f f

A necessary condition that two interior transformations with the same characteristic set (a) be in the same restricted deformation class is accordingly that their invariants ~ be respectively equal. It will be shown in w I2 that this condition is sufficient.

I f (fl) is an admissible reordering of (a), then d , ( f , a) # J i ( f , fl)

in general. I n case of ambiguity we shall refer to J i ( f , a) as the i th invariant Ji with respect to (a).

The value of V(h~) is independent of f and depends on (a). The values of d(h{) obtainable by changing f, differ by integers. One can accordingly include all values of Ji with respect to (a) in the set

(5. I6) Jt = g , ( f o , a) + m,

where mi is an integer and f 0 an interior transformation with the characteristic set (a). W e thus have the theorem,

Theorem 5.3. The invariants J~ (f, a ) f o r a given i belonging to two different transformations f with the same characteristic set (~) differ by an integer mi.

It will be seen t h a t there are functions with the prescribed set (a) for which the integers m~ are prescribed.

We term the sets (J) given by (5. I6) with ms an arbitrary integer the sets (J) associated with (a).

w 6. The existence of at least one interior transformation f with a prescribed characteristic set (a). In this section we shall establish the existence of at least one f with the characteristic set (a) by exhibiting the Riemann image of S with respect to f. In the next section we shall show that except when /~----o and

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m---2, f can be composed with suitably chosen semi-restricted homeomorphisms of S to obtain composite functions f u with invariants J i ( f , a ) + r, where ri is an arbitrary integer. These models are meromorphic only in special cases. Mero- morphic models will be described by formula in w Io.

The following lemma will be used.

L e m m a 6. 1. There exists a sense-preserving homeomorphism T of S onto itself in which the image of an arbitrary point set

(6.

o)

z , . . . , z~

of s distinct points" on S is a prescribed set

( 6 . I ) w , . . ., w ,

of s distinct points on S.

L e t g be a simple arc which joins two points on the boundary of S but which otherwise lies on S and passes t h r o u g h the points (6. o) in the order written.

Let k be a similar are passing t h r o u g h the points (6. I). The closed domains into which g divides S can be carried respectively by sense-preserving homeomorphisms T 1 and T~ into the closed domains into which k divides S and these homeomorphisms can then be modified so as t o transform g into k and in parti- cular to carry the respective points z~ into the corresponding points wi. The lemma follows.

Theorem 6.1. There exists at least one interior transformation f of S with a prescribed characteristic set (a).

The set (a) is proposed as an ordered set of r zeros, s poles and # branch point antecedents.

The function f affirmed to exist will be defined by describing the Riemann image H of S with respect to f over the w-sphere 2~. As defined H will be the homeomorph of S, will cover the point w = o of ~ r times, the point w = c ~ s times, and possess tt simple branch points covering points of ~ distinct from w = o and w--~ c~. One starts with an arbitrary open disc-like piece k of ~ which does not cover w = o or w = o o . One then extends r + s narrow open tongues from k over ~ with tips covering w ---- o r times and w-~oo s times, keeping the e x t e n d e d surface free from branch points and simply connected. To the boundary of k-so extended one joins /~ two-sheeted branch elements making each junction along a short simple boundary arc of the branch element so t h a t . t h e s e elements do n o t cover

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Deformation Classes of Meromorphic Functions. 73 w = o or w---oo. W e can suppose t h a t t h e b o u n d a r y of t h e r e s u l t i n g open R i e m a n n s u r f a c e does n o t c o v e r w = o or w----oo.

T h e R i e m a n n surface H so o b t a i n e d c a n be m a p p e d h o m e o m o r p h i c a l l y in a sense-preserving f a s h i o n o n t o S and, by v i r t u e of the p r e c e d i n g lemma, in such a m a n n e r t h a t t h e r points of H c o v e r i n g w = o, t h e s points c o v e r i n g w - ~ o o , and t h e /~ b r a n c h p o i n t s of H go r e s p e c t i v e l y i n t o t h e p r o p o s e d zeros, poles, a n d b r a n c h p o i n t a n t e c e d e n t s of t h e given set (a).

T h e p r o o f of t h e t h e o r e m is complete.

The case / ~ = o m ~ 2. I n all cases e x c e p t this one t h e c o m p o s i t i o n of a p a r t i c u l a r t r a n s f o r m a t i o n f possessing a p r e s c r i b e d c h a r a c t e r i s t i c set, w i t h a s u i t a b l y c h o s e n h o m e o m o r p h i s m V of S will yield (cf. w 8) a t r a n s f o r m a t i o n f~/

with t h e given set (a) a n d i n v a r i a n t s J~ differing f r o m those of f by a r b i t r a r y integers.

This m e t h o d of c o m p o s i t i o n fails w h e n ~----o a n d m---2. I n this case t h e r e is b u t one f u n c t i o n C~(z,a), n a m e l y C~--= I, so t h a t V(h~) as given by (5.6) r e d u c e s to o. T h e r e is b u t one i n v a r i a n t Ji, n a m e l y J1, a n d

(6.2) = d (h().

T h e set (a) reduces to (a0, al) a n d t h e r e are two cases a c c o r d i n g as al is a zero or pole. I n t h e case of two zeros ao a n d an, t h e only possible values of J~ are

(6.3) 3 i I 3

2 2 2 2

a n d in t h e case of a zero a 0 and a pole a~

(6.4) . . . . 2, - - I , o, I, 2 , ' . . .

W e s t a t e t h e f o l l o w i n g t h e o r e m :

Theorem 6 . 2 . I n ease t * = o and m ~ 2 there exists an interior transformation f of S with a prescribed characteristic set (so, al) with dl arbitrarily chosen from amon 9 the values (6.3) when al is a zero, and f r o m among the values (6.4) when a 1 is a pole.

A p r o o f of this t h e o r e m m a y be given by c o n s t r u c t i n g a R i e m a n n surface f o r t h e i n v e r s e of a f u n c t i o n of t h e r e q u i r e d type. H o w e v e r , t h e t h e o r e m is also established by t h e f o r m u l a s of w Io and t h e more t o p o l o g i c a l p r o o f will be omitted.

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74 Marston Morse and Maurice tteins.

w 7. The variation in J i ( f , a) caused by variation in (a).

L e t f be an interior t r a n s f o r m a t i o n of S w i t h the characteristic set (a). L e t f t , o < t < I, be an admissible d e f o r m a t i o n of f in which (a t) is t h e characteristic set o f f f at t h e t i m e t.

Recall t h a t

(7. o) ,~ (f, a) = d(h 0 -- V (hi),

by definition. As f is deformed, d(h{) remains i n v a r i a n t while

V(hi)

depends on (a) b u t n o t on f l W e have t h e i m p o r t a n t result:

The algebraic increment

(7. i) j j i _~ j , ( f , , al) _ j , ( f o , ao)

in th'e invariants Ji in an admissible deformation f f depends only on the path (a t) and not on f

More explicit f o r m u l a s for 3:~ and d J i are needed. To t h a t end set

(7.2) d(h0 -- ui (rood I),

a n d recall t h a t ui can be t a k e n as ~ when ai is a zero, a n d o when ai is a pole.

L e t ei be i or - - I according as ai is a zero or a pole. T h e n

I I

(7.3) u i - - - - arg e , - (mod I).

2 ~ r 2

To make the f o r m u l a for J~ more definite we shall use a b r a n c h ~ X of t h e a r g u m e n t for which

o ~ a r g X < z ~

signalling this branch by the addition of the bar. I n accordance with the definition of C~ in (5-3)

C,(a,, ,) = B (a~, a) A'(~,_~) (a, -- ao) (i # o) B (ao, a),

Ci(ao, a) = A ' (ao, a) (a~ -- a,).

On r e f e r r i n g to t h e f o r m u l a (5.7)' for

V(hi)

a n d m a k i n g use of (7. o), (7.2) a n d (7.3), one finds t h a t

(7.4)

J~. (f, a)arg[e,A'(a,,a)]

a ~ [A'Lao, a) 1

= t ] -

[B

(ao,-) J + L (i -.:-, . . . . , n)

where Ii is an integer, a n d ei---~ I when a i is a zero, a n d - - I when a,. is a pole.

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Deformation Classes of Meromorphic Functions. 75 The integers l i ( f , a) are invariant under restricted deformations o f f and are uniquely determined by f and (a). They are fundamental in the meromorphic theory.

F r o m (7.4) and the continuity of j ( f t , a t) one obtains the following result:

L e i n m a 7.0. I f ~ , o <= t < i, is an admissible deformation of a transforma- tion f with characteristic set (a) and with (a t) the characteristic set o f f ~, then the difference

J ~ = J~(f', a') -- j , ( f o , ao) (i -= I,. .., n)

(7.5)

is given by (7.6) where

[ ~ t t t t t = l

Oi(Z) = a r g l a ~ , a ~ . ] a r g / a , - - b~t ] J

k = I , . . . , i - - I , i + I , . . . , n; j = I , . . . , ~;

where Z re2resents the path (at); and where the argument in (7.6) is immaterial as long as a branch which varies continuously with t is used.

A p a t h (at), o _--< t =< I, which leads from a set (fl) back to (/3) will be called an a-circuit. W e shall also use paths Z in which (a a) is an admissible reordering of (a ~ = (#). W e t e r m such a p a t h an admissible u.circuit rood ft. As previously, admissibility of a p a t h g requires t h a t a~---ao ~

The difference J J i in (7.5) will concern us not so m u c h as the difference 6j~ = j~(f~, fl) _ j , ( f o , ~) (i = I , . . . , n)

to compare the J~'s with reference to the same set (fl).

because it is necessary W h e n 2 is an a-circuit,

and in this case 6J~ depends only on g and n o t on the initial and final sets (J).

I f g is an admissible u-circuit rood fl which is not an u-circuit, this is never the case, as we shall see.

The group ~ of J-displacement vectors. L e t {d, fl} denote the set of invariants (J) realizable as invariants of an interior t r a n s f o r m a t i o n with the characteristic set (#). I t will presently be seen t h a t the set {J, #} is identical with the complete set of (J)'s ~>associated,> with (fl) in (5.16).

W h e n (at), o _--< t =< I, represents an a-circuit g leading from (#) to (#), the differences J J ~ given by (7.6) are integers ri. I f one sets

Or,. = j f (fo, $) Tz (J~') = J l ( f ' , ,8)

THEIR EXTENSIONS

DEFORMATION CLASSES OF MEROMORPHIC FUNCTIONS AND THEIR EXTENSIONS TO INTERIOR TRANSFORMATIONS.

(2. i) So,

(r > o),

2)

simple,

characteristic.

like character.

admissible.

Admissible fldeformations.

(a t)

restricted.

terminally restricted.

semi-restricted. W e

same X-deformation class

X-equivalent.

The invariants J(. W e

J,(f,

.)

Topological models with prescribed sets (a) and (J).

~,associated>>

restricted

semi-restricted

Meromo~Thic r~odels.

Restricted deformation classes

C'

C"

Terminally restricted deformation classes

category

C'

Semi-restricted deformation classes C".

C"

Other exceptional cases.

Topological and con formal differences.

model sequence.

continuously

covering point

d(k)

w 3. The difference order d(k) of a locally simple arc k.

d(k)

Local simplicity.

w(t) ~- u(t) + iv(t) (o < t <= to)

w(to)

of local simplicity of k.

uniformly

The regular case a ~b.

regular

w(t)

w'(t)

rotational units,

2)

(o _-<

w'(t)

w(t)

(3.

(3.

d(]c) = ~D(]C)_ ~a(]c).

Qa (k, ]ca) Qb (]C, kb)

Qa (k, ]ca, ]cb)

w (t)

(4.2) o=<e<~

A' (ao, ,~)

ao) B (ai, a)

(i > o)

(5.7)'

h', h",

h'

h"

h"

h"

h'

h"

h'

h'

h'

h"

k'

k"

h o

B " : fl(MP)k" (Q N)