This thesis is divided into two parts. The first part uses a family of Interval Exchange Transformations constructed by Michael Keane to show that IETs can have some particular behavior including: (1) IETs can be topologically mixing. (2) A minimal IET can have an ergodic measure with Hausdorff dimension alpha for any alpha ∈ [0,1]. (3) The complement of the generic points for Lebesgue measure in a minimal non-uniquely ergodic IET can have Hausdorff dimension 0. Note that this is a dense Gdelta set.;The second part shows that almost every pair of IETs are different. In particular, the product of almost every pair of IETs is uniquely ergodic. In proving this we show that any sequence of natural numbers of density 1 contains a rigidity sequence for almost every IET, strengthening a result of Veech.
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