Iterated function systems a rising from recursive estimation problems

Summary We show that under suitable conditions the one-step predictor of a finite-state Markov chain from noisy observations has a unique stationary law which is supported by a self-similar set, called the attractor. Under additional symmetry conditions such attractor is either connected, or totally disconnected and perfect. In this latter case the predictor keeps an infinite memory of the past observations. The main problem of interest is to identify those values of the parameters of the chain and the observation process for which this happens. In the binary case, the problem is completely solved. In higher dimension the problem is harder: a complete solution is presented for ternary chains in the completely symmetric persistent case.