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

Abstract

We consider quasi-compact linear operator cocycles ${\mathcal{L}}_{\omega }^n:={\mathcal{L}}_{{\sigma }^{n-1}\omega }\circ \cdots \circ {\mathcal{L}}_{\sigma \omega }\circ {\mathcal{L}}_{\omega }$ driven by an invertible ergodic process $\sigma :\Omega \to \Omega$, and their small perturbations ${\mathcal{L}}_{\omega ,\epsilon }^n$. We prove an abstract $\omega$-wise first-order formula for the leading Lyapunov multipliers. We then consider the situation where ${\mathcal{L}}_{\omega }^n$ is a transfer operator cocycle for a random map cocycle $T_{\omega }^n:=T_{{\sigma }^{n-1}\omega }\circ \cdots \circ T_{\sigma \omega }\circ T_{\omega }$ and the perturbed transfer operators ${\mathcal{L}}_{\omega ,\epsilon }$ are defined by the introduction of small random holes $H_{\omega ,\epsilon }$ in $[0,1]$, creating a random open dynamical system. We obtain a first-order perturbation formula in this setting, which reads ${\lambda }_{\omega ,\epsilon }={\lambda }_{\omega }-{\theta }_{\omega }{\mu }_{\omega }(H_{\omega ,\epsilon })+o({\mu }_{\omega }(H_{\omega ,\epsilon })),$ where ${\mu }_{\omega }$ 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 ${\theta }_{\omega }$. 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.