DIVERGENT SEQUENCES SATISFYING THE LINEAR FRACTIONAL TRANSFORMATONS

A real sequence { x n } 1 ∞ which satisfies the recurrence x n + 1 = a x n + b c x n + d , in which all of 
 a , b , c , d are real will, for certain values of these constants, be divergent. It is the purpose of this 
note to examine the limit 
 lim N → ∞ 1 N ∑ n = 1 N f ( x n ) : f ∈ C ( − ∞ , ∞ ) 
in these cases. Except for certain exceptional values of a , b , c , d this value is found for almost all 
 x 1 .

must allow R to contain the ideal "point at infinity" which we take as o.This will allow the point -e d-to have an image.Thus R =[-oo, + ).) This transformation will have one or two fixed points.In the former case if , is the fixed point, it will be real and any sequence of real numbers satisfying Zn + T(zn) (n 1,2, will converge to ,.If there are two fixed points then either: (a) they will both be real or (b) they will form a complex conjugate pair.In case (a), any sequence generated as above will converge to one of the fixed points (call this one ,) whilst in case (b) there will not be convergence at all.
The behavior of the two convergent cases above can be expressed (more weakly) by asserting that for any !E C( oo, oo).Our purpose here is to enquire about the existence and value of this limit in the non- convergent case (b) above.We shall show that, except for certain exceptional values of a,b,e,d, and for almost all z 1, we shall have whenever/is such that the right hand side exists as a Lebesgue integral.(The constant K is a normalizing constant whose value is such as to make the right hand side unity when I(z) 1.) If and are the two fixed points in case (b) then the transformation can be written as y T(z) az+b c+d 19 this 0 < < 2 since, c being non-zero, the identity transformation is excluded.
Define S: l-[0, 2=) as follows: S(x) arg(Zz--), with the principal value of arg lying in Define H/:[0,2r)--,[0,2r) by HB($)= +$ (rood 2t) Clearly both S and H/ are one-one and onto.It is also well-known that, whenever B is not a rational multiple of , H E is ergodic with respect to ordinary Lebesgue probability measure m on [0,2) and preserves this measure.
When z and y are real (2) is equivalent to That is, cz + d' we see now that T 5' To deal with this situation we shall next prove a lemma (stated in somewhat more general terms than is required for the present application).
LEMMA.Let Ml(Xl,t,m1) and M2(X2,,m2) be two measure spaces with probability measures m and m 2. Let S:Xl-.X2 and H:X2-.X2, both of these transformations being one-one and onto.Suppose that E C X is ml-measurable if and only if S(E)C X 2 is m2-measurable and that ml(E m2(S(E)).Finally, suppose that//is m2-measure preserving and ergodic with respect to this measure.Then the transformation T $-1H$ is ml-measure preserving and ergodic with respect to m 1.
PROOF.Since all our transformations are invertible, we may deal with the transformations themselves rather than their inverses.
(a) Let E c X be rnl-measurable.
(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) so that 0 < m2(B < 1.But, since H is ergodic, this is impossible.Hence there cannot exist any set A satisfying both T(A)= A and 0 < ml(A) < 1.This shows that T is ergodic with respect to the m measure and the proof of the lemma is complete.
We now return to the particular application in hand.As we have alreaAy remarked, when B is not a rational multiple of = the transformation HB of [0,2=) onto itself is ergodic with respect 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.It remains to find m so that ml(E m(S(E)) whenever E C Write _= $(z) ar so that i z--6 By differentiation we find that d -"6 dz (z-o)(z-"6) so that if E C II _= [-,) then Since and "6 are the zeros of cz2+ (d-a)z-b this latter integral can be re-written in an obvious way and so we find that the required ml-measure is given by K dz ml(E)= Ef3[-x,oo) cz 2 +(d_a)z_b with K chosen to make this a probability measure.
Applying the Lemma we see that the transformation T preserves this measure and is ergodic with respect to 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 the extended real axis It onto itself and let cz+d it have a pair of complex conjugate fixed points.If a,b,c,d are such as to make B in (2) not a rational multiple of then, for almost all z (in the m sense *), the sequence defined by ,+b (.= 1,2,...) (d-b,= 1,0) Zn + cz n + d will have the property stated in (1) above.NOTE: *By referring to the definition of m measure above, "almost all in the m sense" can be seen to be equivalent to "almost all in the sense of Lebesgue measure on