Perturbation formulae for quenched random dynamics with applications to open systems and extreme value theory
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We consider quasi-compact linear operator cocycles ℒ^n_ω:=ℒ_σ^n-1ω∘⋯∘ℒ_σω∘ℒ_ω driven by an invertible ergodic process σ:Ω→Ω, and their small perturbations ℒ_ω,ϵ^n. We prove an abstract ω-wise first-order formula for the leading Lyapunov multipliers. We then consider the situation where ℒ_ω^n is a transfer operator cocycle for a random map cocycle T_ω^n:=T_σ^n-1ω∘⋯∘ T_σω∘ T_ω and the perturbed transfer operators 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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