Unique ergodicity and the approximation of attractors and their invariant measures using Ulam's method

Asymptotically stable attractors supporting an invariant measure, for which the ergodic theorem holds almost everywhere with respect to Lebesgue measure, can be approximated by a space discretization procedure called Ulam's method. As an application of this result we propose the use of this method to approximate the `chaotic' attractors of flows in lower dimensions. A Monte Carlo implementation makes this feasible. The approximation method can be extended to attractors whose neighbourhoods contain positively invariant compact sets called blocks. Note that such attractors can fail to have open basins of attraction. When the attractor is uniquely ergodic, we also prove the weak convergence of the approximate measures constructed by the method and as an application, we show the weak convergence of Ulam's method for the logistic map at the Feigenbaum parameter value. More generally, using the work of Buescu and Stewart on transitive attractors of continuous maps, we prove weak convergence of the approximate measures and convergence of their supports to classes of Lyapounov stable attracting Cantor sets.