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An example of an ergodic T(z) ~ = az on K, where a T w h i c h is not w e a k - m i x i n g is not a root of unity. ) (iv) There are examples of w e a k - m i x i n g mixing. T which are not strong- Kakutani has an example c o n s t r u c t e d by c o m b i n a t o r i a l methods, and M a r u y a m a c o n s t r u c t e d an example using Gaussian processes. and K a t o k - S t e p i n also have examples. bility space, let ~(X) if (X,B,m) is a proba- denote the c o l l e c t i o n of all invertible measure-preserving transformations ~(X) Indeed, Chacon with the "weak" t o p o l o g y of (X,8,m).

By p o s i t i v i t y , and hence Therefore UFN(X) + f(x) a max l~n~N f (x) n = max 0ANON f (x) n when FN(X) = FN(X). Thus We have f ~ F N - UF N on A = {x: FN(X) > 0}, so > 0 for 33 IA f d m >- IA FNdm - IA UFNdm = ;X FNdm - ;A UFNdm since FN : 0 >- ;X FNdm - IX UFN dm since FN t 0 = >_ 0 since HUH -< i. on X\A. UF N e 0. 5 hold if U = UT for measure- T. 6: T: X ~ X be measure-preserving. 1 g E LR(m) If i n-i sup ~ [ g(Tm(x)) n~l m=0 B e = {x EX: and > e} then I gdm ~ em(B flA) B NA if and T-IA : A and a m(A) < ®.

N i=0 iff for all space and f (L2(m) 1 iff for all set becomes Then n-i I n i=0 T becomes is a probability ! 9: Suppose (a) of time. independent on the average. The next result Theorem asymptotically, is strong-mixing f,g for all (L2(m), f (L2(m), iff (U~f,g) + (f,l)(1,g) (U~f,f) ~ (f,l)(l,f). 0 is 43 Proof: Ca), Cb), and (c) are proved using similar methods. shall prove (c) to illustrate the ideas. proof will prove (2) = (i). (i) = (3). (XA,I)(I,XB). This follows by putting f =XA, g =XB, for A,B E B.

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