Computable chaotic orbits

We contrast analytic properties of chaotic maps with the results of fixed-precision computation and then use Turing's ideas of computable irrational numbers to illustrate the computation of chaotic orbits to arbitrary N-bit precision. This leads to the study of chaos theory via integer maps that are automata with long-range site interactions. We also explain why the β-shadowing lemma is not a justification for the use of fixed-precision arithmetic in chaos theory.