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.

给定一个由非奇异可测图的共循环表示的离散时间随机动力系统，我们可以通过研究与映射相关的 Perron-Frobenius 算子的共循环来获得有关动力学量的信息。特别令人感兴趣的是算子共环的第二大李雅普诺夫指数λ ₂，它可以告诉我们系统中的混合率和相关性衰减。我们证明了 Banach 空间上有界线性算子的余环的广义 Perron-Frobenius 定理，该空间保持锥体并偶尔收缩锥体；该定理表明，cocycle 的顶部 Oseledets 空间是一维的，并且最大的李雅普诺夫指数λ ₁和λ ₂之间的差距存在下界（即，严格小于λ _{1 的}λ ₂的上限）在与锥收缩相关的量方面明确。然后，我们将这个定理应用于由参数化映射族产生的 Perron-Frobenius 算子的余循环的情况，以获得λ ₂的上限；据我们所知，这项工作是第一次将λ _{2 设定}为一系列地图的上限。为此，我们利用了新的平衡 Lasota-Yorke 不等式。我们还检查了具有两个不变密度的家庭内固定地图的随机扰动，并表明随着扰动缩小到未扰动的地图，λ ₂ 在尺度参数中至少是渐近线性的。我们的估计是尖锐的，因为固定映射的一系列缩放扰动都是马尔可夫，使得λ ₂渐近到尺度参数的 -2 倍。