Time-dependent iteration of random functions

In studying the iteration of random functions, the usual situation is to assume time-homogeneity of the process and some average contractivity condition. In this paper we change both of these conditions by investigating the iteration of time-dependent random functions where all the functions converge (as the iterations proceed) uniformly to the identity. The behaviour of the iterates is remarkably different from the standard contractive situation. In particular, we show that for affine maps in Rd the “chaos game” trajectory converges almost surely. This is in stark contrast to the usual situation where the trajectory moves ergodically throughout the attractor.