Evolution to a singular measure and two sums of Lyapunov exponents

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.