Computer arithmetic and sensitivity of natural measure

For difference equations with bounded, non-periodic solutions, the natural measure of the equation is a useful descriptor of the geometry of the solution. When it exists, it catalogs the density of typical bounded solutions of the equation, and can be used to characterize its dynamics. By their nature, non-periodic solutions evade easy description. Computer methods are usually needed. In case of chaotic solutions, it is futile to try to approximate a particular solution for a large number of steps using floating-point computation, due to sensitive dependence on initial conditions. However, even if individual trajectories are unstable, a compact attractor defined as the totality of a number of solutions may be stable. Given that it is mathematically stable, we ask whether it is computable. In this article we discuss the practical and theoretical obstructions to approximating the natural measure on a computer by creating long trajectories. In particular, we want to know whether the small errors made in computer arithmetic lead to correspondingly small errors in natural measure, or whether they can lead to disproportionately poor estimates. To some extent this is a question about the sensitivity of natural measure to small perturbations, and more precisely the sensitivity to rounding errors made in machine computation. For example, it is clear that near global bifurcation points, natural measure will be sensitive to small changes. We begin by investigating examples of this type, and proceed to an example where there is no nearby bifurcation. In both cases, we find examples of extreme failure of computer arithmetic to approximate natural measure, in that the errors are many orders of magnitude greater than the arithmetic rounding errors.