SPATIAL DISCRETIZATION OF MAPPINGS

State space discretization occurs in the discrete finite arithmetic of a computer. When a dynamical system is simulated by numerical computations, it consequently evolves in a discretized space of this kind. Where attractors are seen in the simulation, what is their relation to the theoretical structures? If a theoretical attractor occurs, should we expect always to see a computational attractor? We address these questions by giving sufficient conditions for a discretized attractor to be present, and show that it converges to the true attractor in the sense of convergence of compact sets in the Hausdorff metric.