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A Problem

Starting with Ulam and von Neumann⁶ and Pitt⁷ there have been various versions of the random ergodic theorem stated and proved. One version is as follows.⁸ Let [X, -, li] be a measure space and [J, A, p] be a probability measure space with J a set of measure-preserving transformations defined on X. Let J* = 11i Ji where i = J for

all i and p* = pli° Pi where pi = p for all i. Then if F(x) e L1(X, t), it follows that {2 I n p* lim -1 f(Tk ... Tx) = f*(x) foralmostall xinX =1 k=1l where (T₁ , T ₂, ...) is a point of J* and f*(x) e L1(X, /).

In the above theorem, the transformations Ti are chosen independently. One can ask, "Suppose the Ti are not chosen independently, but are chosen to form a Markov chain, or are chosen in accordance with the rules governing a P6lya process (see above), does some random ergodic theorem hold?" One could specialize to the case of choosing from two transformations, each of which is a rotation of the circle.