Compound Poisson statistics for dynamical systems via spectral perturbation

Abstract

We consider random transformations $T_ω^n:=T_{σ^{n-1}ω}\circ\cdots\circ T_{σω}\circ T_ω,$ where each map $T_ω$ acts on a complete metrizable space $M$. The randomness comes from an invertible ergodic driving map $σ:Ω\toΩ$ acting on a probability space $(Ω,\mathcal{F},m).$ For a family of random target sets $H_{ω, n}\subset M$ 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 $\mathcal{L}^{n}_{ω,0}:=\mathcal{L}_{σ^{n-1}ω,0}\circ\cdots\circ\mathcal{L}_{σω,0}\circ\mathcal{L}_{ω,0}$, where $\mathcal{L}_{ω,0}$ is the transfer operator for the map $T_ω$. We construct a perturbed cocycle with generator $\mathcal{L}_{ω,n,s}(\cdot):=\mathcal{L}_{ω,0}(\cdot e^{is\mathbb{1}_{H_{ω,n}}})$ and an associated random variable $S_{ω,n,k}(x):=\sum_{j=0}^{k-1}\mathbb{1}_{H_{σ^jω,n}}(T_ω^jx)$, 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 $S_{ω,n,n}$ 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.

Figures (1)

• Figure 1: Graph of a map $T_\omega$, with $\gamma_\omega=2$.

Theorems & Definitions (43)

• Theorem 1
• Theorem 2.1
• Remark 2.2
• Example 2.3
• Example 2.4
• Example 2.5
• Theorem 2.6
• Example 2.7
• Example 2.8
• Definition 3.1