Ergodic Theory on Compact Spaces by Manfred Denker

By Manfred Denker

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Proof: If x ~ OT(X), there exists a neighborhood U such that U A T -n U = ~ for all n ~ I, and hence such that all T -n U, n ~ O, are disjoint. If x were in Supp U, one would have w(T -n U) = u(U) > O, a contradiction to u(X) = I. 7. 1) Definition: Let X be compact metric. ~ Xi, where X i = X i=--~ for all i, endowed with the product topology. X is called the state space. X_l , Xo,Xl,... ) where x i E X. x i is called the i-th coordinate of x. is a compact metric space. If d is a metric on X, one obtains a metric d on X Z by +~ d(x,y) :> 2-1il (xi, Yi) for x,y 6 X Z.

Weakly) mixing with respect to ~, or that the transformation ~ is strongly (resp. weakly) mixing with respect to m. 2) Proposition: A strongly mixing system is weakly mixing; a weakly mixing system is ergodic; the converse statements are not true in general. b). A simple example of an ergodic but not weakly mixing system is given by the permutation on the two point space {xl,x2} with m(Ixll ) = m({x21 ) = 2 -I. Category arguments show that there exist many weakly mixing systems which are not strongly mixing (see [80] and [170]) but it is rather difficult to find examples of this kind: The first one has been constructed by Chacon in [195].

2), lira ~1 ~ K fN o Tj = (fN). = f* in L2(X,~)-norm j=o SO that the last expression tends to 211~i If" Iif* - fN II. Now (*) is equivalent to T,. 1 K-1 lim iim lim ~ 7 - - l i f " M N K j=0 2 , Tj - f M II : 0 and hence to lim lim llfN - f~ 112 = 0. 2). Now let (fn) be a dense sequence in the real C(X) (and O0 hence in L2(X,~j), for any ~). Clearly ET(X) = n~=1 E(fn). Hence ET(X) is a Borel set and ~(ET(X)) = 1. f7 (5,13) Proposition: Let U E ~]~T(X) be ergodic and v E ~(X) be absolutely continuous with respect to ~I.

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