Under what conditions can interval exchanges be approximated by periodic maps? Under what conditions can an interval exchange be approximated by periodic maps? (in the weak topology for the Lebesgue measure on $[0,1]$ ). 
Are there non-trivial examples of periodically approximable interval exchanges (besides circle rotations)
 A: Are you referring to approximation by periodic measurable transformations, or to approximations by periodic transformations with a specific additional structure (such as periodic interval exchange maps)? In any case, as was pointed out elsewhere by Pietro Majer, every interval exchange is a weak limit of interval exchanges with rational endpoints, and such interval exchanges are periodic, so no conditions are required.
However, the condition that a transformation should be an interval exchange is far stronger than is necessary for it to be a weak limit of periodic transformations (without additional structure). Every invertible measure-preserving transformation is such a limit: if $T \colon [0,1] \to [0,1]$ is any invertible measure-preserving transformation then it can be approximated arbitrarily closely in the weak topology by periodic measure-preserving transformations. This result goes back to work of Halmos and Rokhlin in the 1940s: it can be found in Halmos' book Lectures on Ergodic Theory and is a direct consequence of Rokhlin's lemma. For an alternative treatment you could try Steve Alpern's article New Proofs that Weak Mixing is Generic, Inventiones Mathematicae 32 (1976) 263-279. You could view this result as saying that the weak topology does not tell you an awful lot about dynamics: in a way this is intuitive, since knowing that the sets $T_1A$ and $T_2A$ are similar does not help you to understand whether or not $T_1^nA$ and $T_2^nA$ are similar when $n$ is very large.
