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. Such flows appear naturally as special representations of Hamiltonian flows on the torus with critical points. We consider the class of von Neumann flows with one discontinuiny. I will show that any such flow has infinite rank and that the absolute value of the jump of the roof function is a measure theoretic invariant, that is two ergodic von Neumann flows with one discontinuity are not isomorphic if the jumps of the roof functions have different absolute values, regardless of the irrational rotation in the base. The main ingredient in the proofs is a Ranter type property of parabolic divergence of orbits of two nearby points in the flow direction.
Joint work with Adam Kanigowski.