Perturbation formulae for quenched random dynamics with applications to open systems and extreme value theory

We consider quasi-compact linear operator cocycles $\mathcal{L}^{n}_ω:=\mathcal{L}_{σ^{n-1}ω}\circ\cdots\circ\mathcal{L}_{σω}\circ \mathcal{L}_ω$ driven by an invertible ergodic process $σ:Ω\toΩ$, and their small perturbations $\mathcal{L}_{ω,ε}^{n}$. We prove an abstract $ω$-wise first-order formula for the leading Lyapunov multipliers. We then consider the situation where $\mathcal{L}_ω^{n}$ is a transfer operator cocycle for a random map cocycle $T_ω^{n}:=T_{σ^{n-1}ω}\circ\cdots\circ T_{σω}\circ T_ω$ and the perturbed transfer operators $\mathcal{L}_{ω,ε}$ are defined by the introduction of small random holes $H_{ω,ε}$ in $[0,1]$, creating a random open dynamical system. We obtain a first-order perturbation formula in this setting, which reads $λ_{ω,ε}=λ_ω-θ_ωμ_ω(H_{ω,ε})+o(μ_ω(H_{ω,ε})),$ where $μ_ω$ is the unique equivariant random measure (and equilibrium state) for the original closed random dynamics. 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. An extreme value law is derived using the first-order terms $θ_ω$. 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 illustrate the theory with a variety of explicit examples.

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