title: Remark on the (non)convergence of ensemble densities in dynamical systems.
abstract: We consider a dynamical system with state space M, a smooth, compact subset of some 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 Z, T(t)=T(t), or continuous, t in 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 R(n), the expectation value of logrho(x,t), with respect to any stationary (i.e., time invariant) measure nu(dx), is linear in t, nu(logrho(x,t))=nu(logrho(x,0))+Kt. K depends only on nu and vanishes when nu is absolutely continuous with respect to dx.(c) 1998 American Institute of Physics.
authors.affiliation: Departments of Mathematics and Physics, Rutgers University, Piscataway, New Jersey 08854-8019.
authors.forename: S.
authors.initials: S
authors.lastname: Goldstein
authors.forename: J. L.
authors.initials: JL
authors.lastname: Lebowitz
authors.forename: Y.
authors.initials: Y
authors.lastname: Sinai