Remark on the (non)convergence of ensemble densities in dynamical systems

We consider a dynamical system with state space M, a smooth, compact subset of some ℝⁿ, and evolution given by T_t, x_t = T_tx, x ∈ M; T_t is invertible and the time t may be discrete, t ∈ ℤ, T_t = T^t, or continuous, t ∈ ℝ. Here we show that starting with a continuous positive initial probability density ρ(x,0) > 0, with respect to dx, the smooth volume measure induced on M by Lebesgue measure on ℝⁿ, the expectation value of logρ(x,t), with respect to any stationary (i.e., time invariant) measure v(dx), is linear in t, v(logρ(x,t)) = v(logρ(x,0)) + Kt. K depends only on v and vanishes when v is absolutely continuous with respect to dx.