By P. Walters

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**Additional resources for Ergodic Theory: Introductory Lectures**

**Example text**

F ~ L2(m) B 7ioA 2 , .. and by (i) ~ ~" biYi(a)Ti(Ax) is violated. = 1 Yi = ~i biYi(x)" are all distinct then Hence, if yoA = y. V x ~ [a,BG] loT = f, Then bi ~ 0 But then and by (it) then (*) b 1• = 0 T i = I. This follows by the remark above. e. (a) = (d). If So Yi(a) f for = 1 is con- 28 (d) = (c) is t r i v i a l . (c) = (b). , on Tl = i. If Then y(Tx) dense set y ~ i, (ii). G. # g : 7(x) G so a contradiction, T x0 ~ G (i) and Yl(X0), with (ii) of (b) hold. Then = 71(x). {Tnx0 : n z 0} T ~ i, and 3 Yl = yoB.

J + ~Jk+l(n)] has density then n ~ Jk+l and therefore Suppose lanl ~ K V n. < ~i + k-~l . zero. If 1 lanl < ~-~. Hence = 0. such that n ~ Me as n ~ J (2) ~ (I). N + ZJk+l(n)] n ~ Ns, n ~ J ~j(n) - < s. implies imply Then n ~ max(N There and ,M e ) M exists such that implies s jn{o,1 ..... n-l} K < ~ ~j(n) (i) - (3). s > 0. lanl < s~ n K i= 0 fail : n Let + s By the above < i~Jn{0,1 ..... n-l} (K+I)~. lan] = 0 Jinx- iff lira J~n~- Corollary T Z+ fan 12 0. 8: is weak-mixing lim ) = iff m(T-nA nB) V A,B ( B = m(A)m(B) 3 iff J(A,B) V A,B of density zero in ( B n ~ n{J(A,B) 1 n-1 Im(T-iA nB) - m(A)m(B)l 2 ~ 0.

N I(uAf,5) 6 ~ 1 (U~fk,6) I + n : IIf-fkIl2+ I(UAfk,6)l n ~ N(s) where is chosen so that contains G, k l(UXfk,5)I L2(m). Fix (U~f,g) ~ (f,l)(l,g)}. A 7 °Ak : Y (ii) [a,BG] k > 0 Choose then ¢ are isomorphic H6 L2(m). Hence A is on a compact metric abelian iff A is ergodic. is ergodic yeA = y, G is connected. iff and = G. is one-to-one. a unitary operator T implies is ergodic then CT = A¢. ) £f = {g E L2(m): strong-mixing. inequality) < implies it is equal to contains (by the Schwarz is chosen so that n ~ N(s) H8 = Then n llf-fkll2116112 + l(UAfk,5) I if it is clear that BG = G b E G since the endomorphism so that B(b) preserves Haar measure U¢ on as Hilbert L2(m).