Abstract: We consider Markov chains arising from random iteration of functions \(S_{\theta}:X\to X\), \(\theta \in \Theta\), where \(X\) is a Polish space and \(\Theta\) is an arbitrary set of indices. At \(x\in X\), \(\theta\) is sampled from a distribution \(\vartheta_x\) on \(\Theta\), 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.