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, λ2, 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 λ1 and λ2
Nearby in the stack
(that is, an upper bound for
λ2
which is strictly less than
λ1
). 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
λ2
; to the best of our knowledge, this is the first time
λ2
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,
λ2
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