Compound Poisson statistics for dynamical systems via spectral perturbation

We consider random transformations $T_\omega^n:=T_{\sigma^{n-1}\omega}\circ\cdots\circ T_{\sigma\omega}\circ T_\omega,$ where each map $T_{\omega}$ acts on a complete metrizable space $M$. The randomness comes from an invertible ergodic driving map $\sigma:\Omega\to\Omega$ acting on a probability space $(\Omega,\mathcal{F},m).$ For a family of random target sets $H_{\omega, n}\subset M$ that shrink as $n\to\infty$, we consider quenched compound Poisson statistics of returns of random orbits to these random targets. We develop a spectral approach to such statistics: associated with the random map cocycle is a transfer operator cocycle $\mathcal{L}^{n}_{\omega,0}:=\mathcal{L}_{\sigma^{n-1}\omega,0}\circ\cdots\circ\mathcal{L}_{\sigma\omega,0}\circ\mathcal{L}_{\omega,0}$, where $\mathcal{L}_{\omega,0}$ is the transfer operator for the map $T_\omega$. We construct a perturbed cocycle with generator $\mathcal{L}_{\omega,n,s}(\cdot):=\mathcal{L}_{\omega,0}(\cdot e^{is\mathbb{1}_{H_{\omega,n}}})$ and an associated random variable $S_{\omega,n,k}(x):=\sum_{j=0}^{k-1}\mathbb{1}_{H_{\sigma^j\omega,n}}(T_\omega^jx)$, which counts the number of visits to random targets in an orbit of length $k$. Under suitable assumptions, we show that in the $n\to\infty$ limit, the random variables $S_{\omega,n,n}$ converge in distribution to a compound Poisson distributed random variable. We provide several explicit examples for piecewise monotone interval maps in both the deterministic and random settings.