A CLASSICAL ERGODIC PROPERTY FOR IFS - A SIMPLE PROOF

Let {w(i), p(i)} be a contractive iterated function system (IFS) [1, p p. 79-80] with probabilities, i.e. a set of contraction maps w(i) : X --> X with associated probabilities p(i), i = 1, 2,..., N. We provide a simple proof that for almost every address sequence sigma and for al l x the limit lim(n-->infinity) 1/n Sigma(i less than or equal to n) f (w(sigma n) circle w(sigma n-1) circle...circle w(sigma 1) (x)) exists and is equal to integral(X) f(z) d mu(z), where mu is the invariant m easure of the IFS. This is the so called 'ergodic property' for the IF S and was proved by Elton in [3]. However, the uniqueness of the invar iant measure was not previously exploited. This provides considerable simplification to the proof.