Every transformation is disjoint from almost every non-classical exchange

摘要

A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira (Ann Sci école Norm Sup (4) 26(6):645-664, 1993). Recurrent train tracks with a single switch which are called non-classical interval exchanges (Gadre in Ergod Theory Dyn Syst 32(06):1930-1971, 2012), form a subclass of linear involutions without flips. They are analogs of classical interval exchanges, and are first return maps for non-orientable measured foliations associated to quadratic differentials on Riemann surfaces. We show that every transformation is disjoint from almost every irreducible non-classical interval exchange. In the "Appendix", we prove that for almost every pair of quadratic differentials with respect to the Masur-Veech measure, the vertical flows are disjoint.