ON THE APPROXIMATION OF INVARIANT-MEASURES

Given a discrete dynamical system defined by the map-tau: X --> X, the density of the absolutely continuous (a.c.) invariant measure (if it exists) is the fixed point of the Frobenius-Perron operator defined on L1(X). Ulam proposed a numerical method for approximating such densities based on the computation of a fixed point of a matrix approximation of the operator. T. Y. Li proved the convergence of the scheme for expanding maps of the interval. G. Keller and M. Blank extended this result to piecewise expanding maps of the cube in R(n). We show convergence of a variation of Ulam's scheme for maps of the cube for which the Frobenius-Perron operator is quasicompact. We also give sufficient conditions on tau for the existence of a unique fixed point of the matrix approximation, and if the fixed point of the operator is a function of bounded variation, we estimate the convergence rate.