AN ALTERNATIVE ELEMENTARY METHOD FOR APPROXIMATION OF INVARIANT MEASURES FOR RANDOM MAPS

In this paper we describe an alternative elementary method of approximating invariant measures for random maps. Instead of computing Ulam’s matrices associated with the Frobenious-Perron operator for random map we compute matrices which approximate Ulam’s matrices.
Let $T = \{\tau_1(x), \tau_2(x), . . . , \tau_K(x); p_1, p_2, . . . , p_K\}$ be a random map which posses a unique absolutely continuous invariant measure $\hat\mu$ with probability density function $\hat f$. With our elementary method it is possible to develop and implement algorithms for the approximation of the invariant measure $\hat\mu$  with a given bound on the error of the approximation. One of the main advantages of our method is that we do not need to deal with the inverse of the component maps of the random maps. Our result is a generalization of the result of Galatolo and Nisoli (see the paper [12] Galatolo, S. and Nisoli, I, An elementary approch to rigorous approximation of Invariant measures, SIAM J. Appl. Dynamical Systems, 13, No. 2 (2014), 958-985) of single piecewise  expanding maps to results of random maps. We present a numerical example.