Mahler’s conjecture for interval exchange transformations

It is known that almost every interval exchange is uniquely ergodic, but it is not known whether for any smooth curve in interval exchange space, not contained in an affine subspace, almost every point on the curve represents a uniquely ergodic interval exchange. This problem could be considered the “Mahler conjecture for interval exchanges”. The analogous problem in diophantine approximation was stated by Mahler in the 1930’s, proved in the 1960’s and then reproved by Kleinbock and Margulis with a dynamical argument which settled several additional number-theoretic conjectures. We apply a strategy similar to that of Kleinbock and Margulis in the context of interval exchanges.

Work in progress with Yair Minsky.