Parallel Quasi-Monte Carlo Computation of Invariant Measures

where m is the Lebesgue measure of R, has a fixed density f∗. This fixed density gives rise to a physical measure of S, which describes the asymptotic behavior of the chaotic orbits from the statistical viewpoint [3]. Our purpose is efficient computation of such fixed densities. In his book [6], Ulam proposed a piecewise constant approximation method to calculate the fixed density f∗ and conjectured that the resulting piecewise constant densities fn would converge to f ∗. In 1976, Li [4] proved the conjecture for the Lasota-Yorke class of piecewise C and stretching mappings S : [0, 1] → [0, 1]. Since Li’s pioneering work, Ulam’s method has been widely investigated concerning its convergence for various classes of transformations and its error estimates. See, e.g., [5] and the references therein. Although Ulam’s method is practically convergent for almost all the problems encountered so far, it has a shortage, especially for high dimensional transformations. That is, the exact numerical evaluation of a matrix in Ulam’s method be-