Abstract : We consider quasi-compact linear operator cocycles driven by an invertible ergodic process and small perturbations of this cocycle. We prove an abstract pathwise first-order formula for the leading Lyapunov multipliers. This result does not rely on random driving and applies also to sequential dynamics. We then consider the situation where the linear operator cocycle is a weighted transfer operator cocycle induced by a random map cocycle. The perturbed transfer operators are defined by the introduction of small random holes, creating a random open dynamical system. We obtain a first-order perturbation formula for the Lyapunov multipliers in this setting. Our new machinery is then deployed to create a spectral approach for a quenched extreme value theory that considers random dynamics with general ergodic invertible driving, and random observations. Further, in the setting of random piecewise expanding interval maps, we establish the existence of random equilibrium states and conditionally invariant measures for random open systems via a random perturbative approach. Finally we prove quenched statistical limit theorems for random equilibrium states arising from contracting potentials. We will illustrate the theory with some explicit examples.

Keywords : random dynamical system; extreme value theory; open dynamical system, perturbation theory