Fixed points and iterations of mean-type mappings

Let (X, d) be a metric space and T: X -> X a continuous map. If the sequence (T-n)(n epsilon N) of iterates of T is pointwise convergent in X, then for any x epsilon X, the limit mu(T)(x)=(n ->infinity)lim T-n(x) is a fixed point of T. The problem of determining the form of mu(T) leads to the invariance equation mu(T) omicron T = mu(T) , which is difficult to solve in general if the set of fixed points of T is not a singleton. We consider this problem assuming that X = I-p, where I is a real interval, p >= 2 a fixed positive integer and T is the mean-type mapping M =(M-1,center dot center dot center dot,M-p) of I-p. In this paper we give the explicit forms of mu(M) for some classes of mean-type mappings. In particular, the classical Pythagorean harmony proportion can be interpreted as an important invariance equality. Some applications are presented. We show that, in general, the mean-type mappings are not non-expansive.