S. Goldstein, J. L. Lebowitz, Y. Sinai
Remark on the (Non)convergence of Ensemble Densities in Dynamical Systems
(15K, TeX)

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