On rank of von Neumann special flows

TL;DR

This paper proves that certain special flows over circle rotations with a specific type of roof function have infinite rank, indicating complex dynamical behavior and lack of simple approximations.

Contribution

It establishes that special flows over ergodic circle rotations with a $C^1$ roof function and one discontinuity have infinite rank, a new insight into their dynamical complexity.

Findings

Such flows do not have local rank one

They have infinite rank

Implications for dynamical systems theory

Abstract

We prove that special flows over an ergodic rotation of the circle under a $C^1$ roof function with one discontinuity do not have local rank one. In particular, any such flow has infinite rank.