Infinite Compositions of Möbius Transformations

Infinite Compositions of Möbius TransformationsAuthor(s): John GillSource: Transactions of the American Mathematical Society, Vol. 176 (Feb., 1973), pp. 479-487Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/1996221 .

Accessed: 24/11/2014 14:44

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access toTransactions of the American Mathematical Society.

http://www.jstor.org

This content downloaded from 192.231.202.205 on Mon, 24 Nov 2014 14:44:30 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=ams

http://www.jstor.org/stable/1996221?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 176, February 1973

INFINITE COMPOSITIONS OF MOBIUS TRANSFORMATIONS(1)

BY

JOHN GILL

ABSTRACT. A sequence of Mobius transformations It 1_ which converges to a parabolic or elliptic transformation t, may be employed to generate a second

sequence ITf} 1 by setting T = 0-. 0 The convergence behavior of

I.T I is investigated and the ensuing results are shown to apply to continued n fractions which are periodic in the limit.

This paper treats the convergence behavior of sequences of M6bius transfor-

mations ITn(z)I which are generated in the following way:

Let t (z) = (a z + b )AC z + d,), where t = lim tn is either parabolic or elN

liptic. Set T (z) = t (z), Tn(z) = Tn (t n(z)), n =2, 3,

Our approach is essentially the same as that of Magnus and Mandell LI], who

investigated the cases in which the tn and t are hyperbolic or loxodromic, and in

which the tn and t are all elliptic. They established conditions on the fixed

points tunI and lvnj of ltnj that insure behavior of ITn(z)l very much like that ob-

served in the special case tn= t for all n [2]. Convergence is in the extended

plane, so that divergence is of an oscillatory nature only.

The present paper consists of results concerning the two remaining possible

combinations of t and t: n

(1) tn any type and t parabolic, and (2) tn elliptic or loxodromic and t ellip-

tic. The principal result obtained in the investigation of case (2) is an extension

and sharpening of the main theorem in Li].

The parabolic case. First consider the case in which t = lim tn is parabolic,

with a finite fixed point v. Some conditions on the rates at which un and vn ap-

proach v are necessary, as the following example illustrates.

Example 1. Let tn = [nl(n + 1)]Sz + 1, where s = 1 + iy, y A 0. Then t =

z + 1, which is parabolic with fixed point v = m. We have

T (z) = z/(n + W) + Qs),

where Q(s) is the truncated Riemann-Zeta function.

Received by the editors February 1, 1972. AMS (MOS) subject classifications (1970). Primary 40A20.

Key words and phrases. Hyperbolic, loxodromic, elliptic Mobius transformations, parabolic Mobius transformations, continued fractions periodic in the limit.

(1) This research was supported in part by the United States Air Force Office of Sci-

entific Research Grant 70-1922. Copyright 0 1973, American Mathematical Society



480 JOHN GILL [February

It can be shown, L3, p. 2351, that 4(Qs) oscillates finitely as n for the prescribed values of s.

Set X(z)= z/(z - 1). Then X ? t f X(z) = t* (z) and tn(z) are the same

type of transformation Li1] and t* = X 1 0 t 0 X has the fixed point v*= I. Obvi- ous ly

T*(Z) = t o * . * o t*(z) = X ? o T o X(z)

has the same convergence behavior as Tn(z).

Theorem 1. Let tt be a sequence of Mobius transformations converging to a parabolic transformation t, having a finite fixed point v. If there exists an or-

dering of un and vn, the fixed points of tn, such that Iun - vnI and |vn+ 1 - vnl both converge, then the sequence ITn(z)l converges in the extended plane for

every z.

Proof. Assume the tn's and t have been normalized so that andn- bncn= ad -bc = 1, and that a+ d = 2.

First observe that any tn may be written implicitly

(1) 1 (Z) /l = Z- 11 + qni tn(z)-v z- v n

where

kn- = Iif tn is parabolic,

(a c u W)/(a c v if t is nonparabolic n n n n n n n

and

qn = cn if tn is parabolic,

= (kn - 1)/(vn un) if tn is nonparabolic.

It may easily be shown that limkn= 1 and limqn = c 4 0.

Next, set

Yn(z)= -/(z - v), K(z) = k * Z, Q(z) = q + Z.

Then

t (z)= y-1 oQ o K oY(z). n n n n n

Set

w(Z) Q Kn Yo Y Y1(Z), S (Z) =Q K o y (z), n= h, h+ 1, ...

where h will be chosen later. Thus

T(z)= Th 1? Yhlow ? ... 0 W ? S (Z).



1973] INFINITE COMPOSITION OF MOBIUS TRANSFORMATIONS 481

Direct computation shows that w (z) = (pnz + qO)/(r z + 1) where r = n+1 v vn

and p = k + q r Set Wh(z) = w ?. o w (z), and consider the convergence behavior of

1W" b S (Z)} 1 for a fixed value of h. Let Wb(z)= (Ahz + Bb)/(Cbz+ Db), where n n n n n

(2) Ab = pA + r Bb

(3) Bh =q Ab + Bh nl n n - n-i

(4) cb = Pnn 1b + r Db-1

(5) Dh-_ =q c + Db n n n- n-

It follows from (2) and (3) that n

An rlPi + ( -pi)qk k2 +

( Pi) qk 1rkqk3 rk b 1 2 1 2 34

(6) +. + (Hpi)qk rk2 qk2j r k2i

where h < k <... < k_ < h + m = n, 1 < I< 2j. The q? and r-factors alternate, and

(Hp,) designates finite p-products with i> h.

Lemma 1. Suppose Jrh+k 1 are the r-jactors in a term of Ab. Then there Ib are no more than s terms having this specific set of r-jactors in An, where s <

Il1 ki-

Proof. The proof is by induction on the auxiliary recurrence relations:

Abm =Ah + rh Bb and Bb Ah+-1 b + Bb b+ b+m- + +-1bl +- b+m '

We observe that

P2= k.+ qiri= 1 + (vi ui)qi+ qiri

so that, by hypothesis, LIpi converges, and there exists a positive number M such

that both LIp1l and |qi| are less than M for i greater than some h.

Fix e > 0 and choose h so large that the following conditions are met, in ad-

dition to those described above: ln-pi 1| < e/2, for n > h, and m| r I b ~~ ~~~m=1 br+mi < I/M, where I < mintI, M, e (2M + c)}.

Consequently, by the preceding remarks and Lemma 1,

Ab pn < M2( Pi)/M + rI + + E M1p()q/... r/2

< M 2(1I/M) + ***+ Mi + (1( 00 < cI2 .




Hence tAb - 1 llp + /2 <

In an entirely similar manner it may be shown that KCh| < c, for a sufficient-

ly large h.

(2) and (3) give

Ab k+Ab - r b + b b+m b+m-1 qh+mrb+mAb+mI- rb+ b+m-11

from which we obtain

(7) Ah Ab+m 1(kh+m -)A 1+ rh+m h+m ?

Summing both sides of (7),

m ~~~~m b+ m + b+j 1 (8) A b+M Pb E (kb+J - l)A b+j 1 + E r b+j B b+P

j=1 j=1

Upon summing, (3) gives

(9) + Zbq Ab b+m

= qb + qbj b+j-1h j=1

Combine (8) and (9) to obtain

m b

m m b (10) A$b+m = + (k - 1 ++ q h+iAh+i> b Pb +

b+j b+j-1I bi b+ q+

Thus, from (10), if q+n M and A <3,

b+m+l b+ml < 3bkh+m+l Ii + Mlrb+m+lI[l + 3(m+ 2)]

< 3EIkb+m+l II + M(m+ 3)1r b+m+ll

Therefore n

I A -A b I < Z: I Ab - A b+m+n b^+m| -

i= bA+m+j bh+m+j- 1

~ n n

< 3M [t lvb+m+j -

Uh+m+jl + E (m + + 2)|r b l 1

The last expression on the right may be made arbitrarily small by choosing m

sufficiently large and n a positive integer. The Cauchy criterion is satisfied and

we have

(11) lim Ab= I(A, h) - 1. n n-m

Similarly, limn Cb= I(C, h) - 0.

(12) n-oo

It is obvious, from (9), that

(13) lim Bb = n

n --*0o




Also, n -2 n-2 n-2

A'hVh - W'0n = det Wb= 1 (det w;) 1 k. = 1 [1I + q,(v; U)] nn nfl n b

The hypothesis implies the convergence of this product to some number close to one, as n - . Hence

(14) lim (Dl/Bh) = I 0 n-

It is now possible to complete the proof of Theorem 1 for z A v. We have, from (11), (12), (13), and (14),

(Ah/1Bb)S (z) + 1 iim [wh o S Wz]-= limn nn

n'- 0 SZn n n-00 (Ch/Bb)S (z) + (Db/IBb) lh

Thus, limn -0 Tn(z) = Tbh1 ? Yh(/l h)d z A v.

We divide numerator and denominator of Wb o S (v) by S (v) and find, after

some computation, that

limn T (v) = 7 Y(i/l-)

Corollary 1. Let ttnj be a sequence of normalized Mobius transformations converging to t, which is parabolic and has a finite fixed point. If tn(z) (anz + bn)/(cnz + dn), then the convergence of the following four series imply the conver-

gence of I Tn(z)A for every z: 5nj [(an+ 1 + dn+ 1)2 -411, 5njan+ 1 a aI

InIcn+ 1 - cnl, nld n+ I dnl.

The following example shows that the hypotheses of Theorem 1, although sufficient, are not necessary.

Example 2. Let

t (z) = [(v + i)z - v ]/[z + (1 v

where v = Oandv = 1n- I )k/k for n > 2e Then limvn = v =log 2, and

both tn and t are parabolic. An intricate investigation, somewhat similar to the

proof of Theorem 1, shows that 1Tn(z)A converges for every z 7 v.

The elliptic case. We next consider the case in which t = limtn is elliptic.

Theorem 2. Let 1tn} be a sequence of Mobius transformations having fixed points ItunI and tvnr, chosen so that Ikn K 1. Let t = limtn be an elliptic transformation having finite fixed points u and v.

(i) If I un un |1 < o, l|vn - vn1I

< oo, and Hkn ' 0, then 1Tn(z)1 con-

verges for every z except perhaps z = v.

(ii) If IIun - unI <M, ??v n- vn 1 < o, and 1|knj converges, then




ITn(z)1 diverges by oscillation for z / u, v and converges to distinct values for z = u and z = v.

Proof. Set Yn (z) = (z - un)/(z -vn ), Kn(z) = k n Z, w0n (z) = Kn 0 ? Yn-1 0

Y71(z), S (z) = K 0

Y (z), and W0(z) = w * 0.. w0 W (z) = (Ahz +Bb)/(Cbz + Db). n n n n n b n - 1 ~~~~n n n n Then

t (z) = Yn 1 K ? Y0(z)

and

T (z) =Th o Yh n S(z).

As before, w n(z) = (pnz + qn)/(rnz + 1), where pn = kn(vn+

1 - u )/(v7 u

etc.

We choose a positive c and find an h such that -Ab _lJn pil < e and |Cbj < e

for n > h. Thus lim , Bn= I(B, h) _ O and limp,, Db - I(D h) % 1.

The following formula is established by induction:

(1n n 2 /; r b\ 1nrb (15) A n fl Pi ( 1: 1 jjPr 1 m n n- )

h m=h m+1

We observe that 11^npj. = In jkjl ^ ,n ( + sj), where 11sj < o. Therefore, in

case (i), fL1lpij 0, as n . The three terms in (15) tend to zero, as n -

Hence, lim , Ab -0 In similar fashion, lim_ __ Cb = 0.

Consequently,

liin T (z)Th?Yhl TT 0 0 Yl O l(B h) n n n-~~~lm nWh(Sz)) h b- h l ~

(D,hb)

for z A v.

The hypotheses of case (ii), and the observed behavior of the coefficients of h Wb provide a straightforward proof of the next lemma,

Lemma 2. For a fixed z / v, there exist finite numbers M and ho such that

h > h0, n > h, m > h - imply IS (z)I < M and ITb(z) vmI > |u v|/4(1 + M).

Using (1) and the fact that

I I ~~~~v n vn 1 1 V V~~~~~a n+1

t +(z) v '=t (z) v +

(t t (z) - v)(+ I(z) -n v

the following formula may be established by induction on n:

1 ll1nk. n-I m v ml v

(16) n \bh /Tk nm z) vrn+z v

+ ni

(|\k k m1 -I

m=b--I h vm+ um+1

where nh- k. 1 _ 1




We may rewrite (16) in the form

1 (11 k k)(Z - un ______ hib

Tb (z)- v (z v )(V - u) n bh n n

(17) + k (Hk) vm - v1 h h (T(nlI) V+ U+( Z k_ (z) v h 1)

+n-,l kn VM + l- vm + um u M+1 +kh -klh

h+1 h m m )m+V - m+ + vh -

Uh Vh+l 11h+1

Set n n \n

Ilk -exp i o; 1r I k;I

F= F(z) (z - u)/(z - v) (v - u), R = IF| sin(10'|/4), where arg k = 0 =O' (mod 27),

lOt < 77

We choose h so large that the following conditions are satisfied, in addition

to previous stipulations: z- u

(18) 1/1rl K 6 where 1-( ( n

(19) 121 N<-

where h = f2

vh h V+1 -uh+I

n (20) I31 <minjI, R/61F11, where U 1k11 1 + f3'

h

(21) ZVm+ Vm< Rlv-ul h 96(1 + M)

u1<Rlv-. Uj2 (22) u u - 48

(23) l v > IV u m> h-1.

Then, from (17), we obtain

(24) T( -= IFI exp [i (arg F?+ )]?+ -v+H(h n)

n h

where IH(h, n)|< R.

The sum of the first two terms of (24) is a point on a circle C with center

- v) and radius |Fl. Hence 1/(Tn(z) - V) lies in a disc U(h, m) at radius

R with center g on C. R has been chosen so that three tangent discs of radius




R with centers on C can be constructed if the centers of the two end discs are

separated by a central angle of 0'.

Clearly, the sequence I1/(Th(z) - vh)Ih diverges by oscillation, so that

Tn(Z)n h must do likewise. The pattern of divergence bears a close resemblence

to that observed when tn = t for all n. In this special case

(- = I F| exp [i(arg F + n0)] + i1- In'(z) - V u - v

Convergence at z = u is easily established, since Sn(u) 0. We return to

the beginning of the proof of case (ii) and interchange the un's and v Ps, in order

to show convergence at z = v. The development in L[] can be paraphrased to show

that lim Tn(u) 7 lim Tn(v).

Corollary 2. If the transformations tn converge to the elliptic transformation

t, where and - bnc = ad - bc = 1 and Ila - a 11|, YIbn - bn I)' |cn -cnc1, nfl n n n n1 n n1 n and Xjdn - dn 1| all converge, then ITW(z)

(i) converges for z A v, if flk n 0,

(ii) diverges for z A u, v, and converges to distinct values at u and v, if

HI|knI converges.

Continued fractions may be interpreted as compositions of M6bius transfor-

mations, and may be written so as to display the fixed points. Set tn(z)=

unvn/' (- (Un + vn) + z), to obtain

-lI v I liv2 (25) 11 22

(u + vI ) + o(u2 + v2) +...

whose nth approximant is Tn(O). The following two examples are applications of Theorems 1 and 2 to contin-

ued fractions which are periodic in the limit.

Example 3. Let un = unjexp(iOn), vn = vnexp(iqn)5 where limun I

lmlvnl =c 74O, limO On = 0, lim On = 0 0 i (mod 2v7). Then

lim kn limiu /vnlexpLi(On On)]= k = expLi(O - 0)],

so that t is elliptic. Theorem 2, case (i) guarantees the convergence of (25), pro-

vided |unj and |vnj are chosen so that flHun/vnj -* 0, (e.g, unl = 1 - 1l/n

|v - 1 + l/n).

Example 4. Let un = c + ?nc Vn = c + an? where limcn = lim6n = 0, c / 0

Xjn anI < ?? Xnlan+1

I- anI < ??s (e.g., un

= - _ i/n2, v 2 = - + i/n2). Then

t is parabolic, and Theorem 1 insures the convergence of (25).




BIBLIOGRAPHY

1. M. Mandell and A. Magnus, On convergence of sequences of linear fractional

transformations, Math. Z. 115 (1970), 11-17. MR 41 #3621.

2. L. R. Ford, Automorphic functions, McGraw-Hill, New York, 1929.

3. T. J. I 'A.Bromwich, An introduction to the theory of infinite series, Macmillan,

London, 1947.

DEPARTMENT OF MATHEMATICS, COLORADO STATE UNIVERSITY, FORT COLLINS,

COLORADO 80521

Current address: Department of Mathematics, Southern Colorado State College,

Pueblo, Colorado 81005



Documents

Infinite Compositions of Möbius Transformations