Shadowing by computable chaotic orbits

We report new results on the shadowing of computable nonperiodic pseudo-orbits by computable chaotic orbits. While ordinary machine truncation/roundoff decisions can produce only periodic pseudo-orbits, we show how nonperiodic pseudo-orbits of chaotic maps can be generated by using for the truncation decision an algorithm for a computable irrational number. The resulting nonperiodic pseudo-orbits are shadowed by unique chaotic orbits of the dynamical system. We illustrate this by constructing examples of nonperiodic pseudo-orbits along with their unique chaotic shadowing orbits for a hyperbolic system. We conclude that the β-shadowing lemma without additional hypotheses provides no information for inferring chaotic attractor statistics from pseudo-orbit statistics in computation.