  We explore the relationship between Turing completeness and topological
entropy of dynamical systems. We first prove that a natural class of Turing
machines that we call "regular Turing machines" (which includes most of the
examples of universal Turing machines) has positive topological entropy. We
deduce that any Turing complete dynamics with a continuous encoding that
simulates a universal machine in this class is chaotic. This applies to our
previous constructions of Turing complete area-preserving diffeomorphisms of
the disk and 3D stationary Euler flows. The article concludes with an appendix
written by Ville Salo that introduces a method to construct universal Turing
machines that are not regular and have zero topological entropy.
