On a Conjecture of Kůrka. A Turing Machine with No Periodic Configurations

A configuration of a Turing machine is given by a tape content together with a particular state of the machine. Petr Kůrka has conjectured that every Turing machine — when seen as a dynamical system on the space of its configurations — has at least one periodic orbit. In this paper, we provide an explicit counter-example to this conjecture. We also consider counter machines and prove that, in this case, the problem of determining if a given machine has a periodic orbit in configuration space is undecidable.