  Given a discrete-time random dynamical system represented by a cocycle of
non-singular measurable maps, we may obtain information on dynamical quantities
by studying the cocycle of Perron-Frobenius operators associated to the maps.
Of particular interest is the second-largest Lyapunov exponent, $\lambda_2$,
which can tell us about mixing rates and decay of correlations in the system.
We prove a generalized Perron-Frobenius theorem for cocycles of bounded linear
operators on Banach spaces that preserve and occasionally contract a cone; this
theorem shows that the top Oseledets space for the cocycle is one-dimensional,
and there is an readily computed lower bound for the gap between the largest
Lyapunov exponent $\lambda_1$ and $\lambda_2$ (that is, an upper bound for
$\lambda_2$ which is strictly less than $\lambda_1$). We then apply this
theorem to the case of cocycles of Perron-Frobenius operators arising from a
parametrized family of maps to obtain an upper bound on $\lambda_2$; to the
best of our knowledge, this is the first time $\lambda_2$ has been
upper-bounded for a family of maps. To do this, we utilize a new balanced
Lasota-Yorke inequality. We also examine random perturbations of a fixed map
with two invariant densities and show that as the perturbation is scaled back
down to the unperturbed map, $\lambda_2$ is asymptotically linear in the scale
parameter. Our estimates are sharp, in the sense that there is a sequence of
scaled perturbations of the fixed map that are all Markov, such that
$\lambda_2$ is asymptotic to $-2$ times the scale parameter.

Dynamical Systems