Compound Poisson statistics for dynamical systems via spectral perturbation

Abstract

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 ${\mathcal{L}}_{\omega ,0}^n:={\mathcal{L}}_{{\sigma }^{n-1}\omega ,0}\circ \cdots \circ {\mathcal{L}}_{\sigma \omega ,0}\circ {\mathcal{L}}_{\omega ,0}$, where ${\mathcal{L}}_{\omega ,0}$ is the transfer operator for the map $T_{\omega }$. We construct a perturbed cocycle with generator ${\mathcal{L}}_{\omega ,n,s}(\cdot ):={\mathcal{L}}_{\omega ,0}(\cdot e^{is{1}_{H_{\omega ,n}}})$ and an associated random variable $S_{\omega ,n,k}(x):={\sum }_{j=0}^{k-1}{1}_{H_{{\sigma }^j\omega ,n}}(T_{\omega }^{j}x)$, 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.