On special flows over IETs that are not isomorphic to their inverses

TL;DR

This paper establishes a criterion to determine when special flows over ergodic interval exchange transformations are not isomorphic to their inverses, demonstrating that for most such flows with certain roof functions, the flows are not symmetric in time.

Contribution

The paper refines existing criteria for non-isomorphism to the inverse and applies it to a broad class of special flows over IETs with specific roof functions, showing typical non-symmetry.

Findings

Most special flows over IETs with absolutely continuous roof functions and non-zero jumps are not isomorphic to their inverses.

Typical piecewise constant roof functions also produce flows not isomorphic to their inverses.

The criterion provides a new tool for analyzing symmetry properties of special flows over IETs.

Abstract

In this paper we give a criterion for a special flow to be not isomorphic to its inverse which is a refine of a result in \cite{Fr-Ku-Le}. We apply this criterion to special flows $T^f$ built over ergodic interval exchange transformations $T:[0,1)\to[0,1)$ (IETs) and under piecewise absolutely continuous roof functions $f:[0,1)\to\mathbb{R}_+$. We show that for almost every IET $T$ if $f$ is absolutely continuous over exchanged intervals and has non-zero sum of jumps then the special flow $T^f$ is not isomorphic to its inverse. The same conclusion is valid for a typical piecewise constant roof function.