WORDS AND

Internat. J. Math. & Math. Scl.

VOL. 16 NO. 2 (1993) 297-300

297

DIVERGENT

^SEQUENCES

SATISFYING THE LINEAR FRACTIONAL TRANSFORMATONS

A.McD.MERCER

Department

^ofMathematics and Statistics University ofGuelph

Ontario, N1G 2W1, Canada

(Received August

26, 1991 andinrevisedform March31

1992)

ABSTRACT. A

real sequence

{z,}

which satisfies therecurrence

z, + cx.

^+d,+ ^inwhichallof a,b,e,d arereal will, forcertainvalues ofthese constants, bedivergent.

It

isthe purpose of this note toexaminethelimit

Lira f(Zn):

f

^{6_C(-oo, o)}

n=l

in these cases.

Except

for certain exceptional values ofa,b,c,dthis valueis found for almost all

z

KEY WORDS AND PHRASES.

Divergent series, linear fractional transformations, ergodic theory,

Cesaro

means.

1991

AMS SUBJECT CLASSIFICATION CODE.

40A50.

Consider thelinear fractionaltransformation:

T(z)=az+b

_cz+d (ad-bc= l:c#O)

whichis tohavethe propertyof mapping theextendedrealaxisRone-oneonto itself. Itiseasily seenthat anecessaryand sufficient condition for thisis that all ofa,b,c,dbereal.

(Note

^thatwe

must allow R tocontain the ideal "point at infinity" whichwetake as o. This will allowthe point _-e

d-

_{tohaveanimage. ThusR=[-oo,}

+

).)

This transformation will have one or two fixed points.

In

the former case if

,

is the fixed point, it will be real andany sequenceof real numbers satisfying

Zn +

^{T(zn) (n 1,2, will}

converge to

,.

Iftherearetwo fixed points theneither:

(a)

^{theywillbothbereal}

or

(b)

theywillformacomplexconjugate pair.

In

case

(a),

any sequencegeneratedas above willconvergetooneofthe fixed points

(call

^this

one

,)

^whilstincase

(b)

there will not beconvergence^{at all.}

The behavior of the twoconvergent cases abovecan beexpressed

(more weakly)

by asserting that

298 A.M. MERCER N

forany

!

EC( oo,oo).

Our purpose here is to enquire about the existence and value of this limit in the non- convergent ^case

(b)

^{above. Weshall} show that, exceptfor certainexceptional ^{values of}a,b,e,d, and for almost allz_1, weshall have

N

f ⁺

^{oo Kl(z)}

-oocz

2+(d-a)z-b

whenever/is such that the right handside exists asa Lebesgue integral.

(The

constant K isa

normalizingconstantwhosevalueissuchastomake therighthandsideunity when I(z)

1.)

If and arethetwofixed pointsin case

(b)

then thetransformation

canbewritten as

y T(z) _c+d^az+b

19

^{this0< <2since,cbeing}non-zero, the identity transformationisexcluded.

DefineS:l-[0, 2=)^asfollows:

S(x)

arg(Zz--),

^withthe principal value ofarglyingin Define

H/:[0,2r)--,[0,2r)

^by

HB($)=

^{+$(rood 2t)}

Clearly both S and

H/

^{areone-one and onto. It is} also well-known that, whenever

B

^{isnot a}

rationalmultiple of

,

H

E

^{isergodicwithrespect toordinary}Lebesgue probability measuremon [0,2) andpreserves thismeasure.

Whenzandy arereal

(2)

^{isequivalentto}

That is, S(v)

tl#S(z)

or v S

1H/S(z)

So,

sincev ^T(z)=_azcz

+ +

^{d’b we see nowthatT 5’}

1HBS.

To deal with this situation we shall next prove^a lemma

(stated

^{in somewhat more general}

termsthanisrequiredforthe present application).

LEMMA.

Let

Ml(Xl,t,m 1)

^and

M2(X2,,m2)

^{be two measure} spaces with probability measures m and_m2. Let

S:Xl-.X

2and

H:X2-.X2,

both of these transformationsbeingone-one and onto.

Suppose

that ECX is

ml-measurable

^{ifand onlyif S(E)CX}2is

m2-measurable

^and

that

ml(E m2(S(E)).

Finally, supposethat//is

m2-measure

preserving andergodicwith respect tothis measure. Then thetransformation T $-1H$is

ml-measure

preserving andergodicwith respect to_{m 1.}

PROOF.

Since allour transformationsare invertible, we may deal with the transformations themselves rather thantheir inverses.

DIVERGENT SEQUENCES 299

(a) Let

EcX be

rnl-measurable.

Then

ml(E) rn2(S(E)) m2(//$(E)) nl(S- I//S(E)) _rnl(T(E))

This shows that 7"is

ml-measure

preserving.

(b)

Let A be a subset of

X!

^{which is invariant} under T and whose m measure satisfies

0<

ml(A

^<1.

^Let

^{B S(A). Then B is an invariant set under H because}

H(B) H$(A) ST(A) S(A) B. Also

ml(A m2(S(A)) m2(B)

^sothat0<

m2(B

^<1.

^But,

^since

H isergodic, this isimpossible.

Hence

there cannotexist anyset Asatisfyingboth T(A)=A and 0<

ml(A)

^{<1. This shows that T isergodic withrespectto the m measure and theproofof the}

lemmaiscomplete.

Wenow return tothe particular applicationinhand. AswehavealreaAy remarked,when

B

is not arational multiple of

=

^thetransformation

HB

^of[0,2=)onto itselfisergodic withrespect to Lebesgue probability measure m and preserves this measure.

To

apply the

Lemma

we let

Xl_--l,X2--[0,2r),m2=m ^(Lebesgue

probability

measure), H=_H[3

^{and S--the $ of the}

application. Itremainstofindm sothat

ml(E

m(S(E)) whenever E^C Write ^_=$(z) ^{ar so}that ⁱ

z--6

By

differentiationwefindthat

d ^-"6 (z-o)(z-"6)^dz sothat ifECII^_=[-,) then

Since and"6 arethezerosof

cz2+

(d-a)z-b^thislatterintegralcanbere-written inanobvious wayandso wefindthat therequired

ml-measure

^isgivenby

K _dz

ml(E)=

_Ef3[-x,oo)

cz2+(d_a)z_b with Kchosen to make thisaprobabilitymeasure.

Applying ^the Lemma we see that the transformation Tpreserves this measureand is ergodic withrespectto it.

Finally, knowing this, we can apply the Individual Ergodic ^Theorem

[1]

to conclude the following:

THEOREM.

Let T(z)=^az

+

^b_{(ad-bc 1:c 0) map theextendedrealaxisIt}onto itself and let cz+d

it have a pair ofcomplex conjugatefixed points. Ifa,b,c,d are such as to make

B

ⁱⁿ

(2)

^{not a}

rationalmultiple of then,for almost allz

(in

^{them sense}

*),

thesequencedefinedby ,+b (.= 1,2,...) (d-b,= 1,0)

Zn +

_czn

+

^d willhave thepropertystatedin

(1)

^above.

NOTE: *By

referring to the definition ofm measureabove, "almost allinthem

sense"

canbe seentobe equivalent to "almost allinthesenseofLebesguemeasure on

REFERENCES

1.

PARRY,

W.,TopicsinErgodic Theory, CambridgeUniv.

Press,

1981.