Limit theorems for some skew products with mixing base maps

TL;DR

This paper establishes limit theorems such as the central limit theorem for skew product systems with mixing base maps, even when traditional spectral gap conditions are not satisfied, using a novel approach based on random transfer operators.

Contribution

It introduces new limit theorems for skew products with mixing base maps under weaker spectral assumptions than previous methods.

Findings

Proves CLT, local limit, and renewal theorems for skew products.

Results apply to systems with Markov shifts and Young towers.

Extends limit theorems to cases lacking spectral gaps.

Abstract

We obtain central limit theorem, local limit theorems and renewal theorems for stationary processes generated by skew product maps $T(\om,x)=(\te\om,T_\om x)$ together with a $T$-invariant measure, whose base map $\te$ satisfies certain topological and mixing conditions and the maps $T_\om$ on the fibers are certain non-singular distance expanding maps. Our results hold true when $\te$ is either a sufficiently fast mixing Markov shift or a (non-uniform) Young tower with at least one periodic point and polynomial tails. %In fact, our conditions will be satisfied %when $\om$ is the whole orbit the towers. The proofs are based on the random complex Ruelle-Perron-Frobenius theorem from \cite{book} applied with appropriate random transfer operators generated by $T_\om$, together with certain regularity assumptions (as functions of $\om$) of these operators. Limit theorems for deterministic processes whose distributions on the fibers are generated by Markov chains with transition operators satisfying a random version of the Doeblin condition will also be obtained. The main innovation in this paper is that the results hold true even though the spectral theory used in \cite{Aimino} does not seem to be applicable, and the dual of the Koopman operator of $T$ (with respect to the invariant measure) does not seem to have a spectral gap.