A classical ergodic property for IFS: a simple proof

Abstract

Let $\{w_i,p_i\}$ be a contractive iterated function system (IFS) [1, pp. 79–80] with probabilities, i.e. a set of contraction maps $w_i:X\to X$ with associated probabilities $p_i$, $i=1,2,\ldots,N$. We provide a simple proof that for almost every address sequence $\sigma$ and for all $x$ the limit $\lim_{n\to \infty}1/n\sum_{i\le n}f(w_{\sigma_n}\circ w_{\sigma_{n-1}}\circ\cdots\circ w_{\sigma_1}(x))$ exists and is equal to $\int_Xf(z)\,d\mu(z)$, where $\mu$ is the invariant measure of the IFS. This is the so called ‘ergodic property’ for the IFS and was proved by Elton in [3]. However, the uniqueness of the invariant measure was not previously exploited. This provides considerable simplification to the proof.