Singular perturbations of piecewise monotonic maps of the interval

Let t: [0, 1] [0, 1] be a piecewise monotonic, C₂, and expanding map. In computing an orbit {ⁱ(x₀)}_i=0, we model the roundoff error at each iteration by a singular perturbation; i.e.,X_n+1=(X_n)+W, whereW is a random variable taking on discrete values in an interval (-ε, ). The main result proves that this process admits an absolutely continuous invariant measure which approaches the absolutely continuous measure invariant under the deterministic map t as the precision of computation 0.