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

Abstract

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, λ 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 λ 1 and λ 2 (that is, an upper bound for λ 2 which is strictly less than λ 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 λ 2; to the best of our knowledge, this work is the first time λ 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, λ 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 λ 2 is asymptotic to −2 times the scale parameter.