# Every transformation is disjoint from almost every non-classical exchange

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## Abstract

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.

## Keywords

Interval exchange Teichmüller theory Ergodic theory## Mathematics Subject Classification (2010)

37E05 30F60## References

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