On isomorphism problem for von Neumann flows with one discontinuity

Abstract

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.

Slow chaos in surface flows

“…[17]). The absolute continuity of the spectrum is essentially 43 based on a decay of correlations which is square-summable. A recent spectral breakthrough, which goes in the opposite direction, concerns the nature of the spectrum of locally Hamiltonian flows on genus two surfaces, and, to the best of our knowledge, is the first general spectral result for surfaces of higher genus, namely g ≥ 2.…”

“…In the special flows presentation they are flows over irrational rotations under a piecewise linear roof with non zero sum of jumps. For further recent results on von Neumann flows, see[16,43,44].…”

“…In genus one, for a flow with one degenerate singularity, one has a special flow over a rotation, under the roof r (x) = c 0 /x α + c 1 /(1 − x) α for some 0 < α < 1. The assumption in[19] is that α is sufficiently close to 1 43. There are actually technical issues in controlling the part of space where there is not enough shearing.…”