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Given a discrete-time random dynamical system represented by a cocycle ofnon-singular measurable maps, we may obtain information on dynamical quantitiesby 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 linearoperators on Banach spaces that preserve and occasionally contract a cone; thistheorem 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 largestLyapunov exponent $\lambda_1$ and $\lambda_2$ (that is, an upper bound for$\lambda_2$ which is strictly less than $\lambda_1$). We then apply thistheorem to the case of cocycles of Perron-Frobenius operators arising from aparametrized family of maps to obtain an upper bound on $\lambda_2$; to thebest of our knowledge, this is the first time $\lambda_2$ has beenupper-bounded for a family of maps. To do this, we utilize a new balancedLasota-Yorke inequality. We also examine random perturbations of a fixed mapwith two invariant densities and show that as the perturbation is scaled backdown to the unperturbed map, $\lambda_2$ is asymptotically linear in the scaleparameter. Our estimates are sharp, in the sense that there is a sequence ofscaled perturbations of the fixed map that are all Markov, such that$\lambda_2$ is asymptotic to $-2$ times the scale parameter.

Asymptotics for the second-largest Lyapunov exponent for some Perron-Frobenius operator cocycles