Random composing of mappings, small stochastic perturbations and attractors

Abstract

Let T and Hr (0<r≦r0) be continuous mappings of a compact metric space (M,d), d](x,Hr(x))≦r for any x∈M. We consider Markov processes \(\tilde T_r \) with transition functions $$\tilde P^r (x,A) = p\chi _A (T(x)) + q\chi _A (H_r (x))$$ . They are random compositions of T and Hr. We study the existence, uniqueness and asymptotic (r→0) behaviour of \(\tilde T_r \)-invariant measures \(\tilde \mu _r \). We do this by converting the problem into the problem of small stochastic perturbations of the mapping T. The main result is that the weak limit points (for r→0) of the set \(\{ \tilde \mu _r :{\text{ 0 < }}r \leqq r_0 \} \) are measures concentrated on “attractors” of the mapping T.