Coupled iterated function systems that contract on average

We consider iterated function systems (IFSs) acting on a phase space X that, whilst not necessarily uniformly contracting, do satisfy a ‘contraction on average’ condition. We introduce the notion of a coupled IFS acting on a new phase space formed by taking infinite (when X is compact) or finite (when X is not compact) products, in analogy with coupled map lattices. For appropriate couplings, we prove the existence of a unique invariant probability measure for the coupled system and show that it depends continuously on the coupling as the coupling tends to zero. We also prove an ergodic theorem and a central limit theorem for the coupled IFS. The methodology is to introduce a family of transfer operators acting quasi-compactly on an appropriate function space and use results of Keller-Liverani [KL1] to prove continuity of their spectral properties in the perturbation.