Criteria for rank-one transformations to be weakly mixing, and the generating property

We begin with some joint results with Foreman that establish the Borel complexity of the measure preserving homeomorphisms inside the space of measure preserving transformations on the unit circle. We give the relationships between various topologies on the space of measure preserving transformations. We then turn our attention to results in Ergodic Theory. Consider a standard Lebesgue probability space X with an invertible measure preserving transformation T. If T is isomorphic to a transformation obtained by a rank-1 cutting and stacking construction, there is a natural measurable partition of X into the set corresponding to the original interval at the first stage of the construction called O 0 and its complement. The original motivation for this research is to investigate whether the weak mixing property is sufficient to prove that the partition {O0, X O 0} generates. Though we cannot answer this question, we give some specific examples that encourage this notion, and some partial results. We give a construction for a family of non-weak mixing transformations. We then give a sufficient condition for transformations to be weakly mixing.