There exists an interval exchange with a non-ergodic generic measure

TL;DR

This paper demonstrates the existence of an interval exchange transformation with a point whose orbit's distribution converges to a measure that is not ergodic, challenging assumptions about typical behavior.

Contribution

It constructs a specific example of an interval exchange with a non-ergodic generic measure, providing a counterexample to ergodic assumptions.

Findings

Existence of an interval exchange with a non-ergodic generic measure

Orbit equidistributes for a non-ergodic measure in this example

Challenges previous beliefs about typical ergodic behavior in interval exchanges

Abstract

We show that there exists an interval exchange and a point so that the orbit of the point equidistributes for a measure that is not ergodic.