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

TL;DR

This paper develops perturbation formulae for quenched random dynamical systems, enabling analysis of their spectral properties, extreme value laws, and statistical limits under small random perturbations and open system conditions.

Contribution

It introduces a first-order perturbation formula for Lyapunov multipliers and transfer operators in quenched random systems, extending spectral and extreme value theory to open and perturbed dynamics.

Findings

Derived a first-order perturbation formula for Lyapunov multipliers.

Established existence of random equilibrium states and conditionally invariant measures.

Proved quenched statistical limit theorems for random equilibrium states.

Abstract

We consider quasi-compact linear operator cocycles $\mathcal{L}^{n}_\omega:=\mathcal{L}_{\sigma^{n-1}\omega}\circ\cdots\circ\mathcal{L}_{\sigma\omega}\circ \mathcal{L}_{\omega}$ 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.