Commuting point transformations

Summary The question considered is the following: If two invertible measure preserving point transformations commute, in what sense is one a function of the other? The main theorem is the following: If two invertible measure preserving transformations commute, and if the first admits of approximation by periodic transformations, then the second transformation is a piecewise power of the first, where we say that is a piecewise power of if there exists a sequence j(n)] of positive integers such that for each measurable set A the limit of the measure of the symmetric difference of (A) and ^j(n) (A) tends to zero.Research supported in part by NSF grant GP-3752.