ABSTRACT
This chapter provides an efficient numerical algorithm for computing absolutely continuous invariant measures of chaotic discrete dynamical systems. The algorithm applies the idea of the Monte Carlo method to the implementation of Ulam’s original scheme to solve the difficult inverse image problem, especially when the mapping of the dynamical system has complicated expression or the expression is hard to obtain from physical experiments. The chapter examines the idea behind Ulam’s piecewise constant approximations for computing the fixed density of the Frobenius-Perron operator associated with one dimensional mappings of the interval. It shows that the new Monte Carlo approach, which may be called a modified Monte Carlo method, is much better than the original Monte Carlo method.
