2022/06/06 by Jason Atnip, Gary Froyland, Atnip, Jason +5
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2206.02471
openalex publication_date 2022/06/06 · openalex created_date 2022/06/13 · openalex updated_date 2026/07/28
We consider quasi-compact linear operator cocycles Lnω:=Lσn-1ω∘⋯\circLσω∘ Lω driven by an invertible ergodic process σ:Ω→Ω, and their small perturbations Lω,εn. We prove an abstract ω-wise first-order formula for the leading Lyapunov multipliers. We then consider the situation where Lωn is a transfer operator cocycle for a random map cocycle Tωn:=Tσn-1ω∘⋯∘ Tσω∘ Tω and the perturbed transfer operators Lω,ε are defined by the introduction of small random holes Hω,ε in [0,1], creating a random open dynamical system. We obtain a first-order perturbation formula in this setting, which reads λω,ε=λω-θωμω(Hω,ε)+o(μω(Hω,ε)), where μω 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 θω. 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.