Long-time stochasticity for periodic iterations of random surface maps

Abstract

Let $M$ be a closed Riemannian surface, and let $f_1,\ldots,f_s$ be area-preserving $C^2$ diffeomorphisms with fixed positive probabilities such that the resulting random dynamical system is uniformly expanding on average. For each $n$, choose $\omega_1,\ldots,\omega_n$ independently according to these probabilities and set $F=f_{\omega_n}\circ\cdots\circ f_{\omega_1}$. We prove that there exists $\alpha>0$ such that, with probability at least $1-e^{-\alpha n}$, the correlations of any two H\"older observables $\phi,\psi$ under $F^m$ are bounded by $e^{-\alpha n}\|\phi\|_{C^\beta}\|\psi\|_{C^\beta}$, for all $m\leq e^{\alpha n}$. These estimates imply a law of large numbers and a central limit theorem for the Birkhoff sums along $F$ of any zero-mean H\"older observable $\phi$, for exponentially long times in $n$. The limiting Gaussian has variance $\int_M\phi^2\,d\mathrm{vol}$.