Stability and collapse of the Lyapunov spectrum for Perron–Frobenius operator cocycles

In this paper, we study random Blaschke products, acting on the unit circle, and consider cocycle of Perron–Frobenius operators Banach spaces analytic functions an annulus. We completely describe Lyapunov spectrum these cocycles. As a corollary, obtain simple product system where has infinitely many distinct exponents, but arbitrarily small natural perturbations cause complete collapse spectrum, except for exponent 0 associated with absolutely continuous invariant measure. That is, under perturbations, exponents become multiplicity 1, $-\\infty$ infinite multiplicity. This is superficially similar to finite-dimensional phenomenon, discovered by Bochi \[4], that away from uniformly hyperbolic setting, can lead zero. however, its perturbation are explicitly described; further, mechanism quite different. stability cocycles arising general products. give necessary sufficient criterion in terms derivative at fixed point, use show open dense set have final part, prove relationship between single two different spaces, allowing us draw conclusions same $C^r$ function spaces.