  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 \cite{Bochi-thesis}, 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 $C^r$ functions spaces.
