Ergodic theorem for infinite iterated function systems

A set of contraction maps of a metric space is called an iterated function systems. Iterated function systems with condensation can be considered infinite iterated function systems. Infinite iterated function systems on compact metric spaces were studied. Using the properties of Banach limit and uniform contractiveness, it was proved that the random iterating algorithms for infinite iterated function systems on compact metric spaces satisfy ergodicity. So the random iterating algorithms for iterated function systems with condensation satisfy ergodicity, too.