Chapter I. Formal Theory of Invariant Points. SURFACE TRANSFORMATIONS AND THEIR DYNAMICAL APPLICATIONS.

APPLICATIONS.

BY

GEORGE D. BIRKHOFF Of CAMBRIDGE, MASS., U. S. A.

A state of motion in a dynamical system with two degrees of freedom depends on two space and two velocity coiirdinates, and thus m a y be represented b y means of a point in space of four dimensions. When only those motions are considered which correspond to a given value of the energy constant, the points lie in a certain three-dimensional manifold. The motions are given as curves in this manifold. One such curve passes through each point.

Imagine these curves to be cut b y a surface lying in the manifold. As the time increases, a moving point of the manifold describes a half-curve and meets the surface in successive points, P, pr ... In this manner a particular trans- formation of the surface into itself - - namely t h a t which takes a n y point P into the unique corresponding point pr _ is set up.

This fundamental reduction of the dynamical problem to a transformation problem was first effected by POINCAR~, and later, more generally, b y myself?

In order to take further a d v a n t a g e of it I consider such transformations at length in the following paper, which appears here b y the kind invitation of Professors MITTAG-LEFFLER and N6RLU~D. The dynamical applications are made briefly in conclusion. These bear on the difficult questions of integrability, stability, and the classification and interrelation of the various t y p e s of motions.

Chapter I. Formal Theory of Invariant Points.

w I. H y p o t h e s e s .

F o r the present we shall confine attention to the consideration of a one- to-one, direct, analytic transformation T in the vicinity of an invariant point of 1 Dynamical systems with two degrees of~'eedom. Transactions oftheAmerican Mathematical Bociety, vol. I8, r9r 7.

A c t a matheraatiea. 43. Imprim~ le 17 mars 1920. I

2 George D. Birkhoff.

the surface S undergoing transformation. Hence, if u, v be properly t a k e n eoSrdinates with the invariant point at u = v = o , the transformation m a y be written

u~ = au + by + . . . , (I)

v~ -~ cu + dv + . . .,

where the right-hand members are real power series in u, v (i. e. with real coefficients), where u~, v~ are the coSrdinates of the transformed point, and where

(2) ad - - bc > o.

More generally, the n o t a t i o n (u~, vD or Pk ( k = o, • i , • 2 .... ) will stand for the point obtained by applying the k t h iterate (power) of T to (u, v) or P.

F u r t h e r m o r e it will be assumed t h a t there exists a real analytic function Q(u, v), not zero for u - ~ v = o, such t h a t the double integral

, i ~ / Q(u, v)du dv

has the same value when extended over a n y region as over its image under T.

Following a dynamical analogy such a transformation will be called conservative.

Also Q will be termed a quasi-invariant /unction o/ T.

An explicit form for the condition t h a t a quasi-invariant function m u s t satisfy is well-known 1 and m a y be readily derived. If the double integral be expressed in terms of the new variables u~, v~, it takes the form

;/

Q(u,v) Ou Ov

[

Ov Ou

]

~u, Or1 Oul ~ du, d r , ,

,J

where the integration extends over the image of the given region under T.

Since the given region is arbitrary, and since by hypothesis the last written has the same value as I - ] - Q ( u l , v l ) d u l d v t taken over the

integral same region,

we infer t h a t the two integrands are equal. B u t the Jacobian of u, v as to u~, vl is the reciprocal of the Jacobian of u~, vl as to u, v. Hence we obtain

(3)

VOul a v t Ov~ /)ut]

Q ( u ' v ) = Q ( u ~ ' v ' ) L O u ~ v - - O u Or J"

i Cf. E. GOURSAT, Sur les transformations ponctuelles qui conservent les volumes. Bullelin des Sciences MatMmatiques, vol. 52, x917.

Conversely, if Q(u, v) is a real analytic function, n o t zero for u = v = o, and if (3) is true, it follows at once t h a t Q is a quasi-invariant function.

If there exists a second quasi-invariant function Qr not a constant multiple Qr

of Q, it is clear that the ratio ~ is an analytic i n v a r i a n t / u n c t i o n o/ T , not zero for u = v = o. Moreover, if a n y quasi-invariant function be multiplied b y such an invariant function, the p r o d u c t is clearly a quasi-invariant function.

When a conservative transformation T has an analytic invariant function (not a constant), the transformation will be said to be integrable. ~

A transformation T remains conservative under a change of variables, say from u, v to u, v. The quasi-invariant function Q is t h e r e b y modified to a function Q obtained b y multiplying Q b y the J a c o b i a n of u, v as to u, v.

w 2. P r e l i m i n a r y Classification of I n v a r i a n t P o i n t s .

We first make an evident and well-known preliminary classification of in- variant points which is wholly based on the nature of the linear terms in the power series for .ui, %. Under real linear change of variables these first degree terms are transformed among themselves without reference to terms of higher degree. Consequently the t h e o r y of linear transformations applies to these terms.

According to this theory the chssifieation depends largely upon the n a t u r e of the roots of the quadratic equation in q,

q ~ - - ( a + d ) q + a d - - b c ~ o .

In the case at hand this equation is a reciprocal quadratic equation, i. e.

(4)

a d - - bc ~ I.

For, if u ~ v ~ o , we have Q = Q I # o and also 3 u t ~ u I ~ v t 8 vt

Thus from (3) the stated equation (4) follows.

equation will be designated as q and ~ . r

The roots of this r e c i p r o c a l

x It should be observed that the definition refers to the vicinity of an invariant point.

4 George D. Birkhoff.

T h e r e a r e t h e f o l l o w i n g t h r e e cases t o c o n s i d e r . F i r s t , r m a y be r e a l w i t h a n u m e r i c a l v a l u e n o t u n i t y ; T c a n t h e n be t a k e n in t h e n o r m a l f o r m

I.

I U t =QU+ ~ (fmnUmV n,

rn+n=2

( v ~ - - v + ~m,UmV '~.

( e r + z),

W e s u b d i v i d e t h i s case a c c o r d i n g as q is p o s i t i v e (case F) or n e g a t i v e (case I").

S e c o n d l y , ,o m a y b e c o m p l e x a n d so of m o d u l u s I. W i t h this case we g r o u p t h a t case Q = • I in w h i c h t h e t w o e l e m e n t a r y d i v i s o r s a r e d i s t i n c t . H e r e T m a y b e t a k e n in t h e n o r m a l f o r m

ui = u cos 0 - - v sin 0 + ~

% , u " v " , (~ = eY~i~

r e + n - - 2

I I .

v, - - u sin 0 + v cos 0 + ~_~

~p,,,,,u"v".

re+n=2

I t is c o n v e n i e n t t o s u b d i v i d e case I I i n t o t h e

irrational

case I I ' w h e n ~ ^R is

2 I g

i r r a t i o n a l , a n d t h e

rational

cases l I " w h e n 0 = o, a n d I I ' " w h e n O_ _~ _p w i t h -p-

2:~ q q

n o t a n i n t e g e r . Case I I " y i e l d s t h e case Q = i ; a n d I I " , t h e case Q = - - I . T h i r d l y , we h a v e t h a t case in which t h e t w o e l e m e n t a r y d i v i s o r s a r e n o t d i s t i n c t ; h e r e T m a y b e t a k e n in t h e n o r m a l f o r m

I I I .

U 1 ~ "4- U + ~ q)mnUmV n, re+n--2

vl = + v + du + ~ (p,,,, u ' v " ,

m-l-nu2

(e----

+ z),

(d , ' o).

W e s u b d i v i d e this case a c c o r d i n g as q - ~ i (case I I I ' ) or Q = - z (case I I I " ) . I f o n l y l i n e a r t e r m s a r e p r e s e n t in u l , v t we o b t a i n t h e l i n e a r t r a n s f o r m a - t i o n s :

I

I . u , = e u , vl = - v , ( e ~ ~ z ) ,

r

I I . ui = u cos

O--v

sin 0, v~ = u s i n O+v c o s 0 ,

I I I . (d ,= o).

These m a y be regarded as furnishing a first approximation to the corresponding general types. According to our I definition all three linear transformations are conservative with Q = I a quasi-invariant function since areas are left invariant.

F u r t h e r m o r e these cases are integrable with invariant functions u v , u S + v ~, u s respectively.

In the first case a point P will move on a hyperbola u v - ~ const, upon successive application of T or T_I (u, v being taken as rectangular coSrdinates);

in the third case P will move along a pair of parallel lines u ~ const. Unless the point P lies on the degenerate hyperbola u v ~ - o in the first case, or on the pair of coincident straight lines u s - o in the third, P will recede to infinity upon successive application of T or T _ I . When P lies on the degenerate hyper- bola in the first case, it will approach the invariant point (o, o ) u p o n successive application of T or else of T - l , and recede to infinity upon application of the inverse transformation. In the third case all points of the line u --~ o are invariant or are reflected into points of the same line on the other side of (o, o), according as the + or - - sign is used.

On the other hand, in the second case the transformation is a rotation a b o u t (o, o) through an angle 0, and every point P remains at a fixed distance from (o, o) upon successive application of T or T_x.

The essence of the distinction here existing is brought out clearly b y means of the following fundamental definition: if a neighborhood of an invariant point can be so taken that points arbitrarily near the invariant point leave this neigh- b o r h o o d u p o n successive application of T (or of T - l ) , the invariant point is u n s t a b l e ; in the c o n t r a r y case the invariant point is ^stable?

Thus the linear transformations I, I I I are unstable in this sense, while those of t y p e I I are stable.

w 3. An auxiliary Lemma.

Before proceeding to the consideration of formal series for u~, vk (k--~o,

• I, + 2,...), we will establish the following obvious b u t useful lemma:

L e m m a . The linear difference equation of the first order in y(k), y ( k + I ) - ay(]r = c~ k ]c%

x See T. Lzvx-CivxTx, SoFra alcuni criteri di instabilith. Annali di Matematica, Ser. I I I vol. 5, 19ox.

6 George D. Birkhoff.

(a, c, X real, a n d ~ a p o s i t i v e i n t e g e r or zero) a d m i t s a s o l u t i o n 9) (real p o l y n o m i a l in k of d e g r e e ~) if Z@ ~, a n d o t h e r w i s e a s o l u t i o n

s p o l y n o m i a l in k of d e g r e e tt + I).

S u p p o s e first t h a t Z ~ a. L e t us m a k e t h e s u b s t i t u t i o n y = ) k w , w h e n t h e d i f f e r e n c e e q u a t i o n t a k e s t h e f o r m

w(k + ~)-- ~w(k), = ~t, k,.

I f we w r i t e

w ~ w(0) k~ § w(1)k ,~-1 ~- .-- ~- w (~),

we f i n d t h a t w will b e a s o l u t i o n if t h e following c o n d i t i o n s a r e s a t i s f i e d

9 . 9 . 9 . 9 9 9 9 ~ 9 .

On a c c o u n t of t h e a s s u m p t i o n m a d e , w e see a t o n c e t h a t t h e s e e q u a t i o n s d e t e r m i n e r e a l q u a n t i t i e s w(0), w(~) .... , w(~) in succession, a n d l e a d t o a s o l u t i o n of t h e k i n d specified.

I f ), ~ a a s l i g h t l y m o d i f i e d a r g u m e n t a p p l i e s . H e r e we w r i t e y ~ - ~ k w as b e f o r e , a n d t h e n

w - - w (~ ^{k ~'+~ -F} w (1) ]c~ § -.- + w(~+l).

T h e c o n d i t i o n s o n t h e c o e f f i c i e n t s t a k e t h e f o r m (~ + ~)w (~ = ; , (~t + x) ~ w~O) + t~w(~) = o ,

1 . 2

9 . 9 , 9 9 9 9 9 9 9

w (~ -F w (1) -~ 9 9 9 -l- w (~) ~ o.

T h e s e e q u a t i o n s d e t e r m i n e real q u a n t i t i e s w (~ w ( l ) , . . . , w(") in s u c c e s s i o n b u t l e a v e w(, +~) u n d e t e r m i n e d , a l t h o u g h it is t o be t a k e n real.

w 4. F o r m a l Series for u~, v~. Case I.

B y iteration one can obtain convergent series for us, vs in terms of u, v.

In case I the linear terms of these series are evidently ~Su, r respectively.

This fact suggests that higher degree terms m a y be similarly given an explicit form in k, and we shall show this to be the fact.

If u~, v, are real series o/ the /orm I with r (case I~), u~,, vs may be represented /or all integral values o/ k in the /orm

Ilk.

co

us = ~ u + ~ q~),, u'nv ", re+n--2

m + n = 2

where ~),~, ~2(km),, are real polynomials in Qs, Q-s, k o/ degree at most m + n in these variables.

L e t us consider first the quadratic terms in the series for Uk, Vl,.

If in uk, vs we replace u, v b y u~, vl respectively, we obtain u~+~, vk+~ b y definition. B y comparison of coefficients in Ifs above, this leads to the equations

The first three of these equations are obtained b y comparing the coefficients of u ~, uv, v 9 respectively in Uk+l(U, v) and uk(ul, vl); the second three are found b y a like comparison of vk+l(u,v) and v~(ul,v~).

B y considering ~ ) ~ , ~p~) with m + n = 2 as undetermined functions of the index k, it is clear t h a t these six equations constitute six difference equations of the t y p e treated in the lemma of w 3.

Moreover these equations suffice to determine these six functions fully for all integral values of k if their value is known for any particular k. In the case at hand we have of course eD (~ = ~(m~ = o for all m and n, since u0 ~ u, v0 = v.

According to the lemma we can find explicit solutions of these difference equations of a v e r y simple type, namely constant multiples of Qs for the first three equations, and of Q--k for the second three equations. Also the six reduced

8 George D. Birkhoff.

homogeneous e q u a t i o n s o b t a i n e d b y r e m o v i n g the first t e r m on t h e right in t h e six e q u a t i o n s a d m i t the following respective p a r t i c u l a r solutions:

B y a d d i n g real c o n s t a n t multiples of these solutions to t h e respective solutions of t h e n o n - h o m o g e n e o u s e q u a t i o n s , we find a new set of p a r t i c u l a r solutions v a n i s h i n g for k = o as desired.

I n this w a y we o b t a i n t h e explicit values of ~(}1 , ~p~) for m + n = 2:

m ~

(5) t~72 =r176

^{q__Q2 ,}

~i~ )=r162

^{Q - - I '}

~f?2 ~~162162

^{r q-'z ,}

i,/,(k) -- 'P,o(e-~--q 2~) tV~2 -- 'p'~ (e-~--~) 'P~2 = 'Pc, ( r - ~ )

We proceed to s h o w t h a t explicit expressions for to(k) ~ m . , @~. of t h e k)

type

s t a t e d exist also for r e + n = 3 ,

m + n

= 4 . . . . in succession.

To begin with, we write t h e e q u a t i o n s o b t a i n e d b y a c o m p a r i s o n of t h e coefficients of

u ~ v "

in

ve+l(u, v), uk(u~, v,)

a n d

vk+l(u, v), vk(u,,vt) in

t h e respective a b b r e v i a t e d f o r m s :

~(k+l) __ ok ~0 _{f e r n - - ~} g f / l I l + Ore-. q0~ + Pro.,

~p~+') = q - ' qJ,.. + e "~-- ~(~) + Q.,..

T h e expansions c f ekul a n d Q-k vi in u~ (u,, v,) a n d vk(u~, v,) respectively y i e l d t h e first t e r m s on t h e right in these equations. The second t e r m s arise f r o m t h e e x p a n s i o n of

_m,,UW (k) mVln

a n d --m, w(~)

umv"l

-1 in t h e same functions. The last t e r m s arise f r o m t h e e x p a n s i o n of ~o(k) u ~ eft a n d ,0~k~ ~,a Vf respectively, w i t h a + fl < m + n ; " r a f t 1 1 " r a f t 1 1

t h u s Pran a n d

Q,n,,

are 1/near a n d h o m o g e n e o u s in r~(kJ ~ a f , ~ a f ,tj(k~ respectively, w i t h real coefficients, p o l y n o m i a l in r e - a , e l , , , ~p~(.tt + ~< a + fl).

Suppose n o w t h a t we t a k e m + n = 3 a n d a s s u m e t h a t t h e explicit expres- sions for r " r a f t (a + fl -= 2) are s u b s t i t u t e d in

Pmn, Q,~,.

T h e above e q u a t i o n s become l i n e a r difference e q u a t i o n s in w ~k) w ( k ) F u r t h e r m o r e , it is clear t h a t these e q u a t i o n s , t o g e t h e r w i t h t h e f a c t t h a t

~f~)n, W(.~

vanish, d e t e r m i n e these variables c o m p l e t e l y for all integral values of k.

B y a similar process to t h a t e m p l o y e d in t h e case m + n = 2 we m a y a r r i v e n o w a t explicit expressions for ,~(k) ,,,(k) in t h e case m + n---= 3. 'lt~Tt ~t ) "Vr t ~

In this new case we have a non-homogeneous part composed of more than one term. B u t each term is of the form c)~kk ~ occurring on the right-hand side of the equation of the lemma (w 3), since the non-homogeneous p a r t is a polynomial in Q~, Q-~ of degree at most 2.

If we add together the various particular solutions corresponding to each of these terms, as given b y the lemma, we obtain a solution of each difference equation for m § = 3 in the form of a real polynomial in Qk, Q-k, k, of at most the third degree in these variables.

The corresponding homogeneous reduced equation has a solution Q(~-,)k.

If a suitable real constant multiple of this solution is added to the above parti- cular solution of the non-homogeneous equation, a new particular solution is obtained which vanishes for /r ~ o. Solutions of this t y p e are real polynomials in Q~, ~-~, /r of degree at most 3 in these variables, and form the desired expressions.

Proceeding indefinitely in this w a y we establish the truth of the italicized s t a t e m e n t for m § n = 3, m + n ~ 4 . . .

I t is obvious t h a t the coefficients in the polynomials ~ ! n , ~ are them- selves real polynomials in the coefficients of the series u~, v~, save for divisors of the form Q~--Q~ where a and fl are unequal integers.

In the later discussion it is convenient to bring back the case I" (Q < o) to the case I' b y means of the following remark:

If ul, v 1 are real series el the ]orm I with Q < o (case I"), then u2, v2 are o/

the ]orm I' treated above.

w 5. F o r m a l series for u~, v~. Case II.

Next let us consider series of t y p e II in the general case when 0 is incom- mensurable with 2 z .

I / u l , vl are real series o] the /orm I I with 0_0 irrational (case IIf), uk, vk may

2 ~ r

be represented/or all integral values o/ k in the ]orm

uk -- u cos k O - - v sin kO + ~ r f ~ u m v n,

II'k. m+,~-2

vk ---- u sin k ~ + v cos kO + ~ ~( k)~,~ u ~ v ~,

r/t + n ~ 2

where q ~ , ~ are real polynomials in cos kO, sin k~, k o[ degree at most m § n in these variables.

A c t a m a r g a r i t a . 43. Imprim6 le 17 mars 1920. 2

10 George D. Birkhoff.

L e t us i n t r o d u c e new v a r i a b l e s u , v, n a m e l y

~ = u + V - - I v , ~ , = u - - V - - I v .

T h e e q u a t i o n s I I give series f o r ~I, ~ in t e r m s of u , v, which a r e of t h e f o r m I w i t h q = e V ~ 0.

N o w t h e l e m m a of w 3 c a n e v i d e n t l y be e x t e n d e d t o t h e case w h e n a, c, a r e c o m p l e x c o n s t a n t s . H e r e of c o u r s e t h e p o l y n o m i a l f a c t o r s in t h e solutions are no l o n g e r real in general. H e n c e t h e same f o r m a l t r e a t m e n t of ~k, Vk is posssible as was m a d e in case I' f o r uk, vk; in f a c t for t h e case a t h a n d n o n e of t h e divisors q a - - q z are o so t h a t t h e solutions a r e precisely of t h e same f o r m . T h u s uk, vk c a n be e x p r e s s e d as p o w e r series in u, v w i t h coefficients ~ f ~ ,

~ ) of ~ m ~ r e s p e c t i v e l y , p o l y n o m i a l in ,o k, `o-k, k of d e g r e e n o t m o r e t h a n m + n . R e c a l l i n g t h e simple r e l a t i o n b e t w e e n u, v a n d u, v, a n d utilizing t h e t r i g o n o m e t r i c f o r m of qk, `o-k we a r r i v e a t series uk, Vk of t h e d e s i r e d t y p e , s a v e t h a t t h e r e a l i t y of t h e p o l y n o m i a l s ,~(k) ~p~)~ is n o t e s t a b l i s h e d .

A l t h o u g h a n i n s p e c t i o n of t h e a c t u a l f o r m u l a s e m p l o y e d w o u l d establish this r e a l i t y , it suffices t o n o t e t h a t , since uk, Vk are real p o w e r series, t h e real p a r t s of ~p~)~, ~ ) c o n s t i t u t e real p o l y n o m i a l s of t h e t y p e r e q u i r e d .

I n t h e r a t i o n a l case I I , 0 = o , series of t y p e I I are also of t y p e I w i t h q = r. C o n s e q u e n t l y t h e m e t h o d of w 4 leads a t o n c e t o t h e c o n c l u s i o n :

I ] us, vl are real series o/ the ]orm I I with O = o (case I I " ) , ul,, v;, m a y be represented /or all integral values o/ k in the /orm

ll"k.

- - u v ' ,

m§

- - v v " ,

r e + n - - 2

where ~(k~ ~]') are real polynomials in k o / d e g r e e at most m + n - I. 1 T h e r a t i o n a l case 0 ~ o c a n be b r o u g h t b a c k to t h e case O = o:

I / ul, vl are real series o/ the [orm I I with _ 0 = P_ (case II'"), then uq, vq 2 ~ q

are of the /orm I I " .

T h e r e are series similar t o II'k in t h e g e n e r a l r a t i o n a l case, b u t we d o n o t n e e d t o use t h e m .

This fact has been noted by C. L. BouTo~', Bulletin of the American Mathematical Society, vol. 23, I916, p. 73. See also A. A. BESSETT, A case of iteration in several variables, Annals of Mathematics, vol. tT, x9xS--I9x6.

w 5. F o r m a l series for uk, v~. Case I I L Finally we have t o consider case I I I :

I / u~, v~ are real series of the form I I I with Q = i (case I I I ' ) , uk, vk m a y be represented/or all integral values o] k in the /orm

III'k.

I oD

uk = u § ~ ~p(k) u m v'* ,

,g~ mn m-~n~2

m4-n--2

where - m . , w(k) ~P~)n are real polynomials in ]c o/ degree at most 2 m + n - - . i .

We propose to deal with this case by reducing it to the case II" as follows.

Write

and let us make this change of variables in the given transformation. We obtain

u~ v~ ^{= ~} + ~ ep,,,,, u m v m+'*,

rot lr/t 4- n

~ ~ + d ~ + ~ m . u v .

Now the right-hand m e m b e r of each of these equations contains v as a factor.

Hence, dividing the first equation, m e m b e r for member, by the second, we find the equivalent equations

ul ~ u + 2 q)mn umvn'

f, rt 4 - n -- 2

a~

m - i - n ~ 2

which is formally of the type II". Hence by our result in w 5 we m a y write for all integral values of k

12 George D. Birkboff.

Y f T $ g$ ~ '

r e + n = 2

m + n ~ 2

where ~(k)r,,~, ~ are real polynomials in k of degree at most m + n ~ i . Multiplying these two equations together, member for member, we got

m + n ~ 2

where ~(~ is a real polynomial in k of degree a t most m + n - - 2 . Compare this equation with t h a t for uk as a power series in u, v, and so in u, v. The two series must be identical so t h a t the exponent of ~ must be at least as great as t h a t of ~ in every term. Hence ~(k~ vanishes identically for n < m. Consequently,

m n

if we write

"m, m + n )

we have ua expressed in the stated form.

Likewise, if we compare the series for Vk with t h a t for vk, we are led to see t h a t ~ vanishes identically for n < m a n d to write

so t h a t vk is of the stated form.

I t m a y be observed t h a t all of the series employed converge for u, v suf- ficiently small in absolute value. This fact justifies the m e t h o d of formal com- parison employed.

The case I I I with q = - - i is taken care of by the following remark:

I [ u~, vt are real series o/ the [orm 111 with 0 = - x (ease 111"), u2, v~ are o/

the form II".

w 7. Uniqueness o f series for uk, vk.

The following is easily proved:

Lemma. Unless q is a root of unity, a polynomial in Q&, Q-k, k, e l, Q-l, 1 .. . . cannot vanish for all integral values of k, l . . . . , without vanishing identically.

If possible, suppose t h a t the lemma is not true when there is a single variable /r i. e. suppose t h a t there exists a polynomial in Qk, Q--k, k which vanishes for all integral values of k without vanishing identically, although Q is not a root of unity.

In the first place we c a n n o t have ] Q ] > I . F o r in this case divide the hypothetical polynomial b y the highest power of Ck which appears explicitly.

Let /r t a k e on larger and larger integral values. All of the terms of the modified polynomial tend to zero save the t e r m formed by the coefficient of this highest power, inasmuch as Qk becomes infinite more rapidly than a n y power of k. This coefficient is itself a polynomial in k which is not identically o. Hence it cannot approach o as k becomes positively infinite. But, since t h e hypothetical poly- nomial vanishes for all integral It, this is absurd.

The possibility ]Q]< I is disposed of similarly by dividing through by the highest power of ~-k which appears.

Hence we have [r ~ I and m a y write r

~eY=-i 0

where 0 is real. Here we fix upon t h e coefficient of the highest power of k which appears in the hypo- thetical polynomial. An a r g u m e n t like t h a t made above shows t h a t this coef- ficient must approach o as L- becomes infinite through integral values. However, this coefficient is a polynomial in cos k0, sin k0; and - - is irrational since Q is 0

2 7 g

not a root of unity. Hence k0 can be m a d e to differ from an integral multiple of 2 z by nearly a n y assigned q u a n t i t y t for large integral k. Thus tbe coefficient polynomial m u s t vanish when k0 is replaced by the a r b i t r a r y real variable t.

This is impossible.

A similar proof disposes of the case when two or more variables enter.

An application of the lemma shows at once:

The polynomials ~(k)~,,,,, ~p~ o/w167 4, 5, 6 are unique.

I n fact it is clear t h a t t h e difference of two such polynomials with the same subscripts m, n vanishes for all integral k. But these polynomials are of the t y p e dealt with in the lemma, and must therefore coincide.

w 8. T h e f o r m a l g r o u p for T .

The various integral powers of the transformation T combine according to the rule

TkT~ ~ Tk+z,

where k and l are a n y integers whatever.

In the preceding sections we have been led to real

]ormal

series giving Tk

/or all integral values o/ k

in the cases I t, I I 1, II", III', to which all other cases were reduced.

14 George D. Birkhoff.

(6)

The /ormulas

u k ( u . v~) = uk+z(u, v), vk(.z, v~) = vk+t(u, v) hold /or all real values o/ k and 1.

T h e c o n t e n t of this s t a t e m e n t is wholly f o r m a l of course.

I n the eases I I ' , IIIr its t r u t h is a t once obvious. The e q u a t i o n s (6) s t a n d for a n infinite n u m b e r of o r d i n a r y p o l y n o m i a l relations b e t w e e n t h e coefficients

~dk) (P~)~, c~(tl ~ 1 , , c~(k+0, ~p(k+z~ which are k n o w n to hold for all integral values of /c a n d 1. Since these coefficients are t h e m s e l v e s o r d i n a r y p o l y n o m i a l s in k, l, these relations hold identically. Similar reasoning, based on t h e l e m m a of w 7, shows t h a t t h e s t a t e m e n t is also t r u e in cases I ~, IIr.

F r o m t h e italicized s t a t e m e n t t h u s established it a p p e a r s t h a t we h a v e t o deal w i t h a o n e - p a r a m e t e r c o n t i n u o u s g r o u p of f o r m a l t r a n s f o r m a t i o n s a n d t h a t b is a n a d d i t i v e p a r a m e t e r for t h e group. ~ I n t r e a t i n g of its properties we need a few of the general formal ideas for such groups.

W e shall write f o r m a l l y

uk $v Ovk I

(7) ~ u = ~ d k l k = o ' = -5~,k=o"

so t h a t we h a v e t h e following table:

(s)

(I'), $ u = u l o g Q + . - , , Sv = - - v log r (II~), O u ~ - - - O v + . . . , 5 v - ~ O u + . . . ,

(II"), Su~r s + - . . , S v ~ ~P~0u !+ ' ' ' ,

(III'), ~ u = qJ2o--~qDll + ~q~o, u ~ + " ' ' , ~ v = d u + . . . . T h e series (~u, Sv are real f o r m a l power series in u , v.

The series Uk, V~ satis/y the /ormal di//erential equations duk = ~u(u~ vk), ~ - ~ ~v(uk, v~),

(9)

dk '

and the initial conditions Uo ~ u, % = v; conversely Uk, Vk are ]ormaUy determined by these equations and conditions.

C. L. BOUTO~ observed these facts in case II', loc. cir.

To begin with, by differentiating the first equation (6) formally as to /r and noting s y m m e t r y , we find

d d d

u z ( u k .

~ u k ( u . vz)= ~uk+z(u, v)= , vk)

P u t t i n g l = o and recalling the definition of $u we obtain the first of the diffe- rential equations (9). The second equation m a y be deduced in like manner.

The initial conditions u0 ~ u, vo ~ v are clearly satisfied.

Conversely, if we write

uk, Vk

as power series in u, v w i t h o u t constant term and with coefficients which are undetermined functions of /r a n d substitute in the differential equations, we get a t each step linear differential equations of the first order in these coefficients. When joined with the condition t h a t all of these coefficients are o for k ~ o , save the coefficients of u in u0 a n d of v in v 0 which are i , these equations successively determine the coefficients.

These facts explain the complete analogy between the classification of trans- formations T near an invariant point and the classification of differential equa- tions of t y p e (9) at a point $u = ~v = o. This analogy was noted by PoI~cAR$. ~

w 9. T h e i n v a r i a n t o p e r a t o r

L(w).

We shall now define the i n v a r i a n t operator L(w):

(~o) L(w) ~ u ~ + ~VS~v .~

I t is clear the

L(w(u, v))

is the formal derivative of w(uk, vk) as to k for k = o.

Consequently L(w) is unaltered (formally) b y a change of variables. The fund- a m e n t a l p r o p e r t y of this operator is expressed in the following s t a t e m e n t :

The necessary and su[/icient condition that a [ormal series F be invariant under T is that L ( F ) = o.

First, this condition is necessary. In fact, if F is an i n v a r i a n t series we have

F(u~, vk)= F(u, v)

for all integral values of /~. Hence, by the lemma of 1 Sur les courbes ddfinies 2ar les dquations diffdrentielles, Journal de mathdmatiques, ser. 3, vols. 7--8, t 8 8 t - - i 8 8 2 a n d ser. 4, vols. i - - ~ , x885--i886. T h e a n a l o g y was e x p l a i n e d p a r t i a l l y b y m e a r t s of a l i m i t i n g p r o c e s s b y S. Lx~T~s, Sur les 4quations fonc~ionelles qui d~finissent une courbe ou une surface invariante par une transformation, Annali di Matematica, ser. 3, vol. x3, i9o 7.

This is t h e *symbol of t h e i n f i n i t e s i m a l t r a n s f o r m a t i o n m i n t h e t e r m i n o l o g y of LI~.

16 George D. Birkhoff.

w 7, this relation holds for all values of k.

k ~ o, we find L ( F ) = o.

Secondly, this condition is sufficient.

Differentiating as to /r and t a k i n g F o r if L ( F ) = o we find, using (9),

d OF(ua, vk)dul, OF(us, va)dva

~-fcF(ua, va) Oul, d k + Ova d k

~- L ( F ( u a , va)) = o.

Hence we infer t h a t F(uk, Vk) is a power series with coefficients independent of k. P u t t i n g k = o we get F(ua, vD = F ( u , v), a n d in particular F ( u l , vi) F ( u , v). T h a t is, F is i n v a r i a n t u n d e r T.

w ~o. E x i s t e n c e of i n v a r i a n t series.

I n w167 2-- 9 the fact t h a t T was assumed conservative did n o t enter, save t h a t we made use of the equation (4). We shall now prove the following:

A n y conservative trans/ormation T o/ the /orm I r, I I r, I I " or IIIr leaves in- variant a real /ormal series F* de/ined by the equations

0 F* 0 F*

(Is) Ov = Q ~ u ' ~ - - - Q ~ v .

B y multiplying together the equation (3) for u, v, for u = ul, v = vz . . . for u = u a - 1 , v = va-1, we obtain

Oua Oua Ou Ov

(3a) Q(u, v) = Q(ua, va)

Ova Ova Ou Ov

for a n y positive integral value of k. We employ the familiar rule for the combination of Jacobians in obtaining this result. Likewise (3k) holds for k---o and also for negative integral values of k, as is easily seen.

Hence this relation (3k) will hold identically when the formal'series for us, va are substituted. This follows from the lemma of w 7.

Differentiating with respect to k and se~ting k = o, we find

Here we have employed the definitions (7) of 6u, $v and we have made use of the fact that the Jacobian determinant reduces to

II~ I

[o i

for k----o. Now consider the terms of a particular degree in

Q6u

and - - Q ~ v . These homogeneous polynomials p and q have the property

Op Oq OU Or'

deduced from (I2). Hence there exists a homogeneous polynomial r of degree one higher such that

Or Or _{- - ,}

P = O v ' q = o u

The sum of the polynomials r of all degrees ( > 2 ) is the formal series F*

required.

From the equations (ii) we have immediately

L(F*)----o,

so that by w 9 the series F* is formally invariant under T.

If a change of variables from

u, v

to

U, V

be made, the series F* for the new variables can be obtained by direct substitution. For, from the equations (~i) we find

OF*OU OF*OV [Ou Ou ]

O---if- O-v + O--V O~ = Q [.O U O U + -O-V 6 V ,

OF*OU OF*OV COV 6 u Ov ]

o--U o~--7 + o~V o ~ - = - q [U6 + U~o v2 9

Ov Ou

Multiplying the first of these equations by ~ , and the second by ~-~, and adding, we find

OF*

^r0u 0v 0u 0vl

But the quasi-invariant function for the new variables is the product of

Q(u, v)

and the Jaeobian of u, v as to U, V (w i). Hence the equation last written shows that F*(u, v), regarded as a formal series in U, V, satisfies the first equation (zx) for the new variables. Similarly the second equation (ix) is seen to hold in these variables.

A c t a mathematlca. 43. Imprim6 le 18 mare 1920, 3

18 George D. Birkhoff.

F r o m the equations (8) immediately evident:

a n d ( i i ) the explicit forms of the series F* are (I'), F * = u v l o g e + . . . ,

(II'), F * = - - 0 ( u ' + v ~ ) + . . - ,

2

(II"), F* -- - - ~ ~/2o uq + " " , (III'), F* = - - d - u ' + -.-.

2

I t is apparent t h a t a n y formal power series in F* furnishes an i n v a r i a n t series.

In order to determine to what e x t e n t the existence of formally invariant series for a transformation I', II', II", I I I ' is characteristic of conservative trans- formations we need to make a digression.

w i i . F a e t o r i z a t i o n o f f o r m a l series. 1

We consider formal series w i t h o u t constant terms. Such a series will be called p r i m e when it cannot be expressed as the product of two others. Since the lowest degree of a n y term in a p r o d u c t is the sum of the lowest degrees for a n y terms in the factors, a n y formal series can be decomposed into prime factors in at least one w a y , and the number of such factors c a n n o t exceed the degree of the initial terms of t h a t series.

Two factors, either of which can be obtained from the other by multipli- cation b y a formal series with c o n s t a n t term, are regarded as essentially equi- valent. Since products and quotients of formal series with constant terms yield series of the same type, the p r o p r i e t y of this convention is obvious.

B y a linear change of variables a n y series G ( u , v) can be given the form c v ~ + . . . , c ~ o, where the indicated terms are of degree at least n. A n y pos- sible factor of G is readily seen to have the same prepared form. Also Wv.IV.R- STRASS's faetorization theorem holds formally, i. e., we m a y write G = E H where E is a power series with constant term c a n d H is a power series, v n + -.., in which v does n o t occur with an exponent as large as n after the first term.

_1

Now let us determine the formal series 8 ( u -~) in powers of u ~ which satisfy the equation H = o, and let us proceed at each step of this determination pre-

i Cf. W. ~'. OSeOOD, Factorization of analytic functions of several variables, Annals of Mathe- matics, v o l . z9, Z9iT--X918.

cisely as t h o u g h H ( u , v) were a polynomial in u , v. T h e well-known m e t h o d for doing so yields higher and higher t e r m s of such series, with ~ a ~ n .

At first sight it might seem conceivable t h a t this process breaks down a t some p o i n t so t h a t it is n o t possible to proceed further. But, since the process used involves o n l y a finite set of t e r m s of H a t each stage, t h e same difficulty would necessarily arise if H were broken off a t some a d v a n c e d term. This is absurd since t h e n we are dealing with a polynomial. Thus we obtain a c o n t r a - diction. Consequently we can o b t a i n formal series of t h e s t a t e d t y p e in u s which, when s u b s t i t u t e d for v, reduce H to o. T h e initial terms in these power series are a t least of t h e first degree in u .

L e t ~ be a n y a t h r o o t of I a n d consider

This p r o d u c t is precisely H , a t least if H is a polynomial in u as well as in v.

B y breaking off H a t an a d v a n c e d t e r m a n d employing a limiting process, we infer t h a t the same is always true.

The b r a c k e t e d p r o d u c t s involve only integral powers of u as well as of v, a n d are prime factors of G. Indeed, if such a p r o d u c t P is n o t prime, its c o m p o n e n t factors are of p r e p a r e d form a n d m a y be decomposed as G has been. B u t a n y new series S so o b t a i n e d must fail to reduce P to o when we write v ~ S . This is absurd.

F o r a similar reason it appears that, if a prime series divides a p r o d u c t , t h e series m u s t divide one of t h e factors.

Ib follows t h a t , as /ar as the /undamental theorems o/decomposition are con- cerned, the situation /or convergent series carries over directly to divergent series.

w i2. Condition for conservativeness.

We are now in a position to p r o v e the following:

A necessary and su//icient condition that a trans/ormation T given by real series I r, I I r, I I ' , I I I ~ (but otherwise unrestricted) be conservative is (z) that there exists a real invariant series F o/ lowest terms one degree higher than those o/

8u, ~v and containing each common prime /actor o/ ~u, ~v to precisely one power higher than it appears as a common /actor in ~u, ~v, and (z) that the /ormal power series given by the equal ratios

20 George D. Birkhoff.

OF OF

Ov Ou

u ' ~v conver~Tes.

Before entering upon the proof, it m a y be observed t h a t an inspection of

~ u , ~ v as given b y (8) shows that, in the cases I r, ][I r, ~u and ~v have no common factor. In these cases the condition (I) reduces to the condition merely t h a t there exists a formal series F with lowest terms of the second degree. I t will appear later that ~u and ~v admit of a common factor only in the extra- ordinarily special cases I I ' , IIIr when there exist curves through (o, o) made up of invariant points.

We first prove the conditions necessary.

We take F = F*. The equations (ii) show t h a t this invariant series has lowest terms of degree one higher than the terms in ~u, ~v of least degree, inasmuch as Q possesses a constant term.

F r o m the equations (iI) it follows also t h a t the ratio series of the italic- ized s t a t e m e n t converges to Q. I t remains to show t h a t F * contains the common prime factors of ~u, ~v to a power one higher than these occur as common factors of ~u, ~v.

L e t pk be the highest power of a n y such prime P occurring in 3u, ~v.

B y (II) we have

OF* p1,a, OF*

Ou 8-~ = Pkb

where either a or b is prime to P .

If F* contains P to higher than the (k + i ) t h power, ~ - ~F* and - - contain P to higher than the kth power.

the equations last written.

If F* contains P to a power m F* ~ PING, we find

OP p OG = pk+l_,,~a, mG uu + ~u

OF*

~-v

will This is in manifest contradiction with with o < m < k + I , and if we write

0 G = p1,+l_,,~b"

m G ~ + P ~ v

0 P OP

Hence, since G is prime to P , both ~ and ~ are divisible b y P . At least

one of these partial derivatives is possessed of initial terms of lower degree t h a n P , so t h a t this possibility is likewise excluded.

The s t a t e m e n t under consideration is certainly true then unless, perchance, F* is n o t divisible by the prime factor P . We have merely to eliminate this possibility.

I t was seen in the preceding section t h a t we can write

CO

when E is a formal power series with constant term, where S in an ascending power series in it, s argument, and where ~o stands for a n y n t h root of I .

Now introduce the variable t ~ u ~ instead of u. We have OF* ntn_ 1 ~F*

OF* e? F* OF*

while ~ is unaltered. Hence the Partial derivatives ~ a n d ~ are divis- ible by v m S ( t ) . L e t us effect a f u r t h e r change of variables from v, t to w, z where w ~ v w S(t), z = t. E v i d e n t l y one has

OF* 8 F* 8 F* OF* O F* d S Ow i)v ' 8z O~ + ~)v dt

OF* OF*

so t h a t ~ and ~ - are divisible by w.

The fact t h a t - ~ - is divisible by w shows t h a t F* contains no terms in z 0F*

alone and is divisible b y w.

Passing back to the variables v, t, we i n f e r t h a t F* expressed as a power series in v, t is divisible b y v - - S ( t ) . I t follows t h a t F * ( u , v) is divisible by v-S(u ) and by P of course. his oompletes the proof that the conditions s t a t e d are necessary.

I t remains to prove t h e m 8u]]icient.

We m a y assume t h a t an i n v a r i a n t series F* exists for which (II) holds in which Q is a convergent power series with constant term I. These equations follow at once from the second part of the italicized s t a t e m e n t under considera- tion. Our aim is to show t h a t T is conservative.

22 George D. Birkhoff.

B y direct differentiation and use of the formal differential equations (9) we obtain

IJ

/ ~ Q (uk, vk) ~ u (uk,

IOUk OUl,]

OQ(uk, vk) dV(Uk, vk) +

+ ~)vl, 8 Vk 0 Vk

I

+

+ Q(uk, vk) Ova 8vk O~v(uk, vD 8~v(uk, vk)

~u 8v ~)u 8v

B u t the first d e t e r m i n a n t in the final brace is the Jacobian of

~u(uk, vk). Vk

with respect to u, v. This determinant m a y be broken up into the product of

O u(uk, vk) I

the Jacobian of

~U(Uk, Vk), Vl:

as to

uk, Vk

which is ~ ! and the J a c o b i a n of

uk, v~

as to u, v. Likewise the second determinant in the same brace m a y

&~v(uk, Vk)

and the Jacobian of u~, vk as to u, v.

be expressed as the p r o d u c t of

~Vk

Hence we find t h a t the right-hand member of the above equations reduces to

I ^{O +}

IOuk Ouk

O@k[Q(uk' ^{Vk) ~Vk]} Ovk Ovk I-g-uu ov

The first factor vanishes identically b y (ii). Hence the left-hand member of the a b o v e equation vanishes identically in k. Integrating formally we obtain (3~), F o r k ~ I this becomes (3), which is precisely the condition t h a t T be conservative with a quasi-invariant function Q.

I t is natural to call a transformation T of types I', II t, I [ ~, IIIt

]ormally conservative

if there exists a formal series F satisfying the conditions in p a r t (I) of the italicized statement.

We m a y inquire precisely w h a t condition the existence of formally in- variant series lays upon transformations T of these types. The ratio Q of the

italicized s t a t e m e n t m a y or m a y not be convergent. If it is convergent, then t t Q ( u , v ) d u d v is i n v a r i a n t under T. If the ratio is not convergent, the double integral is only formally invariant.

These considerations bring out the vitally close connection between conserv- ativeness and formally i n v a r i a n t series.

w ~ 3 . T h e f o r m a l v a n i s h i n g o f t h e J a c o b i a n .

To complete our t r e a t m e n t of formally i n v a r i a n t series we need to establish the formal extension of a well-known property of Jacobians:

The Jacobian o~ two /ormal series in u, v without constant terms vanishes identically i/ and only i/ either can be expressed as a power series in the other or in /ractional powers o/ the other. 1

I t is immediately a p p a r e n t t h a t , if two functions A, B are so expressible one in terms of the other, their Jacobian will vanish identically.

Suppose, conversely, t h a t A a n d B are power series in u, v with vanishing Jacobian:

0A 8B ~A 8B

~)u ~v ~v ~u o.

B o t h A and B are exact powers of base series for which it suffices to establish the functional relation. B u t the Jacobian for the bases also vanishes. Conse- q u e n t l y we m a y confine a t t e n t i o n to the case in which neither A nor B is an exact power other t h a n the first.

We begin by showing t h a t A and B have the same prime factors.

If this is not the case, suppose t h a t A is divisible by a prime series P , while B is not. After a suitable preliminary change of variables, P is expres- sible as a p r o d u c t of series v - - S ( u ~) (w ii). Now take new variables

S ( 1 )

W ~ V - U ~ ₉ t ~ U ~ .

The series A and B are power series in these variables w i t h o u t constant terms, and their Jacobian as to w, t is o b y direct reckoning:

The presence of fractional powers means that the root indicated is to be formally extracted.

24 George D. Birkhoff.

O A O B ~?AOB i)w Ot Yt Ow o.

B u t A is divisible by w, and ~ OA is divisible by w to a power at least as high.

Also ~ OA is divisible by w to a power at least one lower t h a n A. Hence O-B-Bot is divisible by w. From this it follows t h a t B is divisible by w.

Proceeding to the original variables we infer t h a t B is divisible by the prime factor P , c o n t r a r y to hypothesis.

Suppose t h a t a prime factor P is contained p times in A and q times in B, and choose t h a t factor for which p- ~ o is as small as possible, and thus smaller

q

t h a n for some other factor unless p- is the same throughout. Except in this q

case, ~ Aq will yield a power series w i t h o u t constant term a n d n o t containing P . B u t the Jacobian of this series and A is easily verified to be o also. This is not possible by the a r g u m e n t used above, since B~ has n o t the prime factor P Aq which A admits.

We are thus forced to the conclusion t h a t the power series ~ starts off Aq with a constant term. B u t A and B are not exact powers so t h a t we m u s t have p ~ q . Consequently the prime factors of A a n d B occur with the same multiplicity in A and B.

Now consider

A = B(c + C), (c ~ o),

where U is a power series without constant term. I t is readily inferred t h a t

_1 C q .

the Jacobian of C, B is o, a n d thence t h a t , if C is an exact qth power, ~ is a power series with constant term. Hence we m a y write

C = Bq (d + D ) , (d ~ o),

where D is a power series without c o n s t a n t term.

definitely we find

A = c B + d B q + ....

Proceeding in this w a y in-

This establishes t h e statement.

w I4. T h e t o t a l i t y o f i n v a r i a n t series.

We m a y now prove the following:

1/ F* is a qth power the most general invariant series is an arbitrary power series in F*q. The integer q is I unless all the prime /actors o/ F* are common to ~u, ~v.

The results of w I3 assure us t h a t the most general invariant series can be represented as s t a t e d if the Jacobian of F* and a n y invariant series F vanishes.

B u t we have L ( F * ) - - - o , L ( F ) - - - o , whence it appears that the J a e o b i a n does vanish.

If q ~ i we m a y write F * = Gq, and (ix) gives

OG OG Q~v,

qGq-l~v ~ Q J u , qGq-li) u

so t h a t all of the factors of G (and hence of F*) are common to 3u and ~v.

w x5. Conditions for Formal Conservativeness.

At the very outset of the paper the condition (4) was obtained as a conse- quence of the fact that T was assumed to be conservative. There exist an in- finite set of similar conditions on the coefficients of higher degree terms in the power series ul and vl. These conditions m a y be found b y use of the existence of invariant formal series. We illustrate the method in case I r.

Since F* begins with a term u v log t~ in this case, an invariant series F , also with first degree term u v l o g e, can be written down without a n y other terms having equal exponents in u, v:

00

F = uv log Q + ~ F~nu'nv '~, (m ~ n).

m-l-n--3

This series F m a y be obtained b y writing F - ~ F * + c F *~ + ..., and choosing the arbitrary coefficients so as to eliminate terms with equal exponents.

Moreover, it is easy to see t h a t there is only one such series, since a n y invariant series can be expressed as in a power series in F * (w I4).

Now, when coefficients of umv n are compared, the formal relation F ( u l , v,) F ( u , v) gives a series of equations

A c t a raathcmatiea. 43. Imprim6 le 18 mars 1920. 4

26 George D. Birkhoff.

F m , ( e ~ n - ' - 1) = P m n , (m + n > 3 ) . H e r e Pan is a linear e x p r e s s i o n in t h e q u a n t i t i e s Faa w i t h a + fl < m + n . T h u s we d e t e r m i n e Finn for m + n = 3 , m + n = 4 . . . as p o l y n o m i a l s in t h e coeffi- c i e n t s ~m~, ~vm~ of t h e series f o r ut, yr. F o r m = n we h a v e P , , , - - - - o .

I n the case I f the polynomials P,,,, in ~0,~, ~ , ~ (c~ + fl < 2n) vanish/or n =

2~ 3~ . . . .

C o n v e r s e l y , if t h e s e v a n i s h we h a v e a f o r m a l l y i n v a r i a n t series F , a n d f o r m a l c o n s e r v a t i v e n e s s of T in c o n s e q u e n c e .

Similar conditions /or formal conservativeness can be ]ound in the other cases.

w

i6. I n v a r i a n t f o r m a l c u r v e s .

L e t ] a n d g be t w o f o r m a l p o w e r series in a p a r a m e t e r t, w i t h o u t c o n s t a n t t e r m s a n d n o t b o t h i d e n t i c a l l y o. T h e n we shall r e g a r d t h e e q u a t i o n s

u = / ( 0 , v = g ( t ) ,

as f u r n i s h i n g a /ormal curve through the point (o, o). If t h e series /, g c o n v e r g e f o r ]t I small we h a v e a n a n a l y t i c c u r v e .

T w o c u r v e s of this s o r t will be r e g a r d e d as i d e n t i c a l if o n e can be o b t a i n e d f r o m t h e o t h e r b y c h a n g e of p a r a m e t e r t = l(v) w h e r e l is a f o r m a l p o w e r series in ~ or a f r a c t i o n a l p o w e r t h e r e o f .

A f o r m a l c u r v e is r e g a r d e d as real if t h e coefficients in / a n d g c a n be t a k e n real.

B y m e a n s of T a f o r m a l c u r v e of this s o r t is r e g a r d e d as c a r r i e d o v e r i n t o t h e f o r m a l c u r v e

u = u , ( l ( t ) , g ( t ) ) , v = v , ( l ( t ) , g ( t ) ) .

I f this t r a n s f o r m e d c u r v e is i d e n t i c a l w i t h t h e g i v e n c u r v e u = / ( t ) , v - ~ g ( t ) t h e n t h e g i v e n c u r v e is said to be /ormally invariant u n d e r T .

T h e d e t e r m i n a t i o n of t h e f o r m a l l y i n v a r i a n t c u r v e s is essential for o u r p u r p o s e . A f u n d a m e n t a l division of t y p e s of i n v a r i a n t p o i n t s will be m a d e a c c o r d i n g as t h e r e do or d o n o t exist c u r v e s of this s o r t g i v e n b y real series.

I n cases I r, IF, I I ' , I I F t h e t r a n s f o r m a t i o n T will be called hyperbolic if real f o r m a l l y i n v a r i a n t c u r v e s exist, a n d elliptic in t h e c o n t r a r y case. In eases I I "r or I l l " , T is hyperbolic or elliptic a c c o r d i n g as T~ or T2 (of t y p e I I ' ) is o n e or t h e o t h e r .

If tl denotes the power series in t or a fractional power thereof along the transformed invariant curve which relates its p a r a m e t e r and t, we have

/ ( t , ) - - u , ( / ( O , g ( t ) ) ,

g(t,)

= v , ( / ( t ) ,

g(t)).

In virtue of the fact that the determinant of the coefficients of the first degree terms in u,, % is not o (see (4)) we can show that the power series tl starts off with a first degree term in t. For suppose it commences with a term of higher degree. The initial term of one of the two right-hand members above will be a, where a is the lowest degree of a n y term in / or g. B u t the left-hand members will start off with higher degree terms, which is impossible. Similarly we m a y rule out the possibility t h a t the initial term in t is of lower degree than the first, b y making use of the inverse equations

/(t)

= U_l(l(t,), g(t,)), g(t) = v_i(l(t,), g(t,)).

Hence tl is a power series in t or a fractional power thereof beginning with a term of the first degree.

If a is the degree of the lowest term in / or g (say in ]), then from the corresponding equation (the first) we obtain on the left a series in t 1, a t l + . . . , and on the right a similar series in t commencing with a term of degree not less than a and therefore of degree precisely a b y the above. E x t r a c t i n g ath roots we conclude finally t h a t tl can be expressed as an ordinary power series in t with first degree term:

t~ -~ q * t + . . . .

H a v i n g this explicit form of t in mind, let us compare anew the two members of each of the pair of equations first written. We write

l ( t ) = v t ~ + ' ' ' , g ( t ) = q t ~ + . . . ,

so t h a t I P ] + ] q I r o, and obtain

pq*'~ = a p + b q , qr = c p + d q .

I t follows at once that Q*~ is a root of the characteristic equation, i. e. t h a t r r

If (o, o) is an 'ordinary point' of the formal curve we have a = I , r B y successive transformation of the invariant curve b y T , we obtain not only t, b u t parameters t2, t~ . . . Likewise b y the inverse transformation we obtain parameters t_~, t _ 2 , . . . . These can all be obtained from the series for t~

b y iteration.

28 George D. Birkhoff.

w ~7. T h e f o r m a l s e r i e s for tk and t h e f o r m a l g r o u p .

Since the constant Q* is an a t h root of Q, it is clear that, if we write v~(v) = t~(v), then we have v~ = Q r + .--. B y iteration vk m a y be defined for all integral values of k. Moreover, the methods used in w 4 serve at once to show that

(k) m

where eft) is a polynomial in Qk of degree at most m if Q ~ i, and a polynomial in k of degree at most m - - z if r I.

For all integral values of k and l we have obviously

Therefore, b y the lemma of w 7, this holds formally for all real values of k and 1.

We write

dvki

and can then show (compare with w 8) t h a t the formal differential equation

dk

is satisfied, and, together with the initial condition v 0 = v , wholly determines the series for Vk.

w ~8. The invariant operator

L(u, v).

We shall define a second invariant differential operator:

(z4)

L(u, v ) = ~ u d v - - ~ v d u .

I t can be immediately verified that, if the variables

u, v

are changed to u, v, then

L(u, v)

becomes L(~, ~) multiplied b y the Jacobian of ~, ~ as to u, v.

I t is also obvious that, if

u = / ( t ) , v----g(t)

is a formal curve, then

L(u, v)

is independent of the particular p a r a m e t e r chosen for the curve.

The necessary and su]/icient condition /or the invariance el a ]ormal curve

u ~ / ( t ) , v = g ( t ) under T is L ( u , v ) = o .

By definition of invariance we have for such an invariant curve

l(t,)

⁼

u,(l(t), g(t)), g(t,)-- v,(t(t), g(t)),

a n d thence for integral values of k

/(tk) = u k ( / ( t ) , g(O), g(t~) = vk(/(t), g(t)).

If we take k as an integral multiple kra of a (w I7) and write t ~ , ~ l ~ t ~ ( ~ ) (w i7), we have in particular

/(~k,) = .k,~(t(~), g(~)), g(~k,) = vk,.l/(~), aO:)),

for integral values of k r.

L e t the general series for uk,,, Vk,~, vk,, be substituted in the last equations.

All the coefficients are either polynomials in r q--k,, k F (case It), or in cos k'O, sin krO, k r (case IIr), or in k' (cases I I ' , III'). Hence, by the lemma of w 7, these equations are identically true from a formal standpoint.

Differentiating formally as to k' and setting k ' = o, we get

d ! ~ = . ~ u ( / , g), ~ = . ~ v ( / , g),

whence at once L ( u , v ) - ~ o.

Conversely, let us assume t h a t L ( u , v) is o for a formal curve u----/(t), v ~ g(t), and let us show that. the curve is i n v a r i a n t under T.

In this case we have

where ~ is the sum of a polynomial in ~ a n d a power series in t. Now, since I

6u, ~v begin with terms of the first degree or of higher degree, both right-hand members have initial terms of degree at least as high as / or g. On the other h a n d d / d-t a n d s / are of degree one less t h a n t and g respectively. I-Ience z(t) dg cannot contain negative powers of t or even a constant term. Thus • is an ordinary power series in t without c o n s t a n t term.