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

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 for the cocycle of operators, 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 a lower bound for the gap between the largest Lyapunov exponents lambda (1) and lambda (2) (that is, an upper bound for lambda (2) which is strictly less than lambda (1)) explicitly in terms of quantities related to cone contraction. 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 work is the first time lambda (2) has been upper-bounded for a family of maps. In doing so, we utilize a new balanced Lasota-Yorke inequality. We also examine random perturbations of a fixed map within the family with two invariant densities and show that as the perturbation is scaled back down to the unperturbed map, lambda (2) is at least 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.