  We consider dissipative dynamical systems represented by a smooth
compressible flow in a finite domain. The density evolves according to the
continuity (Liouville) equation. For a general, non-degenerate flow the result
of the infinite time evolution of an initially smooth density is a singular
measure. We give a condition for the non-degeneracy which allows to decide for
a given flow whether the infinite time limit is singular. The condition uses a
Green-Kubo type formula for the space-averaged sum of forward and
backward-in-time Lyapunov exponents. We discuss how the sums determine the
fluctuations of the entropy production rate in the SRB state and give examples
of computation of the sums for certain velocity fields.
