On isomorphism problem for von Neumann flows with one discontinuity

TL;DR

This paper proves that for von Neumann flows with one discontinuity, the absolute value of the slope of the roof function is a measure-theoretic invariant, distinguishing non-isomorphic flows regardless of the base rotation.

Contribution

It establishes that the slope's absolute value uniquely determines the isomorphism class among von Neumann flows with one discontinuity.

Findings

Absolute slope value is a measure-theoretic invariant.

Two flows with different slope absolute values are not isomorphic.

The result holds regardless of the irrational rotation in the base.

Abstract

A von Neumann flow is a special flow over an irrational rotation of the circle and under a piecewise $C^1$ roof function with a non-zero sum of jumps. We prove that the absolute value of the slope is a (measure theoretic) invariant in the class of von Neumann special flows with one discontinuity, i.e. two ergodic von Neumann flows with one discontinuity are not isomorphic if the slopes of the roof functions have different absolute values, regardless of the irrational rotation in the base.