Abstract
Let t: [0, 1] → [0, 1] be a piecewise monotonic, C2, and expanding map. In computing an orbit {τ i(x 0)} ∞ i=0 , we model the roundoff error at each iteration by a singular perturbation; i.e.,X n+1=τ(X n )+W ɛ , whereW ɛ is a random variable taking on discrete values in an interval (-ε, 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 ε → 0.
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Boyarsky, A. Singular perturbations of piecewise monotonic maps of the interval. J Stat Phys 48, 561–569 (1987). https://doi.org/10.1007/BF01019688
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DOI: https://doi.org/10.1007/BF01019688