2023/08/21 by Atnip, Jason, Froyland, Gary, González-Tokman, Cecilia +1
#Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2308.10798
We consider random transformations Tωn:=Tσn-1ω∘⋯∘ Tσω∘ Tω, where each map Tω 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 Hω, 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 Lnω,0:=Lσn-1ω,0∘⋯\circLσω,0\circLω,0, where Lω,0 is the transfer operator for the map Tω. We construct a perturbed cocycle with generator Lω,n,s(⋅):=Lω,0(⋅ e^is\mathbb1_Hω,n) and an associated random variable Sω,n,k(x):=∑j=0k-1\mathbb1_Hσjω,n(Tωjx), 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 Sω,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.