Abstract: We consider Markov chains arising from random iteration of functions $S_{\theta }:X\to X$, $\theta \in \mathrm{\Theta }$, where $X$ is a Polish space and $\mathrm{\Theta }$ is an arbitrary set of indices. At $x\in X$, $\theta$ is sampled from a distribution ${\vartheta }_x$ on $\mathrm{\Theta }$, and the ${\vartheta }_x$ are different for different $x$. Exponential convergence to a unique invariant measure is proved. This result is applied to the case of random affine transformations on ${\mathbb{R}}^d$, giving the existence of exponentially attractive perpetuities with place dependent probabilities.

2010 AMS Mathematics Subject Classification: Primary 60J05; Secondary 37A25.

Keywords and phrases: random iteration of functions, exponential convergence, invariant measure, perpetuities.