Compound Poisson statistics for dynamical systems via spectral perturbation

We consider random transformations T ω n := T σ n − 1 ω ∘ ⋯ ∘ T σ ω ∘ 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 → Ω acting on a probability space ( Ω , F , m ) . For a family of random target sets H ω , n ⊂ M that shrink as n → ∞ , 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 L ω , 0 n := L σ n − 1 ω , 0 ∘ ⋯ ∘ L σ ω , 0 ∘ L ω , 0 , where L ω , 0 is the transfer operator for the map T ω . We construct a perturbed cocycle with generator L ω , n , s ( ⋅ ) := L ω , 0 ( ⋅ e i s 𝟙 H ω , n ) and an associated random variable S ω , n , k ( x ) := ∑ j = 0 k − 1 𝟙 H σ j ω , n ( T ω 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 → ∞ 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.