Every transformation is disjoint from almost every non-classical exchange

TL;DR

This paper demonstrates that almost every non-classical interval exchange transformation is disjoint from every other, extending classical results to a broader class of dynamical systems related to quadratic differentials.

Contribution

It establishes disjointness properties for non-classical interval exchanges and their relation to quadratic differentials, generalizing classical interval exchange results.

Findings

Every transformation is disjoint from almost every irreducible non-classical interval exchange.

Almost every pair of quadratic differentials has disjoint vertical flows.

The results extend classical disjointness properties to non-classical interval exchanges.

Abstract

A natural generalization of interval exchange maps are linear involutions, first introduced by Danthony and Nogueira. Recurrent train tracks with a single switch which we call non-classical interval exchanges, 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. 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.