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

Given a discrete-time random dynamical system represented by cocycle of non-singular measurable maps, we may obtain information on quantities studying the Perron–Frobenius operators associated to maps. Of particular interest is second-largest Lyapunov exponent for operators, λ2, which can tell us about mixing rates and decay correlations in system. We prove generalized theorem cocycles bounded linear Banach spaces that preserve occasionally contract cone; this shows top Oseledets space one-dimensional, there lower bound gap between largest exponents λ1 λ2 (that is, an upper strictly less than λ1) explicitly terms related cone contraction. then apply case arising from parametrized family maps λ2; best our knowledge, work first time has been upper-bounded In doing so, utilize new balanced Lasota–Yorke inequality. also examine perturbations fixed map within with two invariant densities show as perturbation scaled back down unperturbed map, at least asymptotically scale parameter. Our estimates are sharp, sense sequence all Markov, such asymptotic −2 times