In this paper, we study random Blaschke products, acting on the unit circle, and consider the cocycle of Perron–Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. As a corollary, we obtain a simple random Blaschke product system where the Perron–Frobenius cocycle has infinitely many distinct Lyapunov exponents, but where arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum, except for the exponent 0 associated with the absolutely continuous invariant measure. That is, under perturbations, the Lyapunov exponents become 0 with multiplicity 1, and $-\infty$ with infinite multiplicity. This is superficially similar to the finite-dimensional phenomenon, discovered by Bochi [4], that away from the uniformly hyperbolic setting, small perturbations can lead to a collapse of the Lyapunov spectrum to zero. In this paper, however, the cocycle and its perturbation are explicitly described; and further, the mechanism for collapse is quite different.

We study stability of the Perron–Frobenius cocycles arising from general random Blaschke products. We give a necessary and sufficient criterion for stability of the Lyapunov spectrum in terms of the derivative of the random Blaschke product at its random fixed point, and use this to show that an open dense set of Blaschke product cocycles have hyperbolic Perron–Frobenius cocycles.

In the final part, we prove a relationship between the Lyapunov spectrum of a single cocycle acting on two different Banach spaces, allowing us to draw conclusions for the same cocycles acting on $C^r$ function spaces.

在本文中，我们研究作用于单位圆的随机 Blaschke 乘积，并考虑作用于环上解析函数的 Banach 空间的 Perron-Frobenius 算子的余循环。我们完整地描述了这些共环的李雅普诺夫谱。作为推论，我们获得了一个简单的随机 Blaschke 乘积系统，其中 Perron-Frobenius 共循环具有无限多个不同的 Lyapunov 指数，但任意小的自然扰动会导致 Lyapunov 谱完全崩溃，除了与绝对连续的指数 0 相关的指数 0不变测度。也就是说，在扰动下，Lyapunov 指数变为 0，多重性为 1，$-\infty$ 具有无限多重性。这表面上类似于 Bochi [4] 发现的有限维现象，远离均匀双曲线设置，小的扰动会导致李雅普诺夫谱崩溃为零。然而，在本文中，明确地描述了共循环及其扰动；而且，崩溃的机制也大不相同。

我们研究了由一般随机 Blaschke 产品产生的 Perron-Frobenius 共环的稳定性。我们根据随机 Blaschke 乘积在其随机不动点处的导数给出了 Lyapunov 谱稳定性的必要且充分的标准，并用它来证明 Blaschke 乘积的开稠密集具有双曲线 Perron-Frobenius 共环。

在最后一部分，我们证明了作用于两个不同 Banach 空间的单个余环的李雅普诺夫谱之间的关系，从而使我们能够得出作用于 $C^r$ 函数空间的相同余环的结论。