A von Neumann flow is a special flow over an irrational
rotation of the
circle and under a piecewise C1 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.