Ergodic theory is concerned with the extent to which iterates of a function mix subsets of a measure space. I will use the following basic concepts of ergodic theory specialized to the measure space with the measure . A measure-preserving function is ergodic if the only -measurable sets E for which are and the empty set, i.e., such a function does such a good job of mixing points in the space that it has no nontrivial -invariant sets. It can be shown [[39], p. 36] that an equivalent condition for ergodicity is that

for all and all integers . This condition in turn is equivalent to the assertion that for almost all the sequence of iterates

is uniformly distributed for all . A function is strongly mixing if

for all and all . Strongly mixing functions are ergodic.
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