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)


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.

  • $\begingroup$ Approximation by periodic measurable transformations, each preserving a finite measurable partition. $\endgroup$ – Mostafa Oct 8 '13 at 22:42
  • $\begingroup$ Ok, then the above answer applies. No conditions are necessary, and indeed the result applies to all invertible measure-preserving transformations of a Lebesgue space, not just to interval exchanges. $\endgroup$ – Ian Morris Oct 9 '13 at 8:40

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