  For a sequence of sub-additive potentials, Dai [Optimal state points of the
sub-additive ergodic theorem, Nonlinearity, 24 (2011), 1565-1573] gave a method
of choosing state points with negative growth rates for an ergodic dynamical
system. This paper generalizes Dai's result to the non-ergodic case, and proves
that under some mild additional hypothesis, one can choose points with negative
growth rates from a positive Lebesgue measure set, even if the system does not
preserve any measure that is absolutely continuous with respect to Lebesgue
measure.
