Quasi-periodic configurations and undecidable dynamics for tilings, infinite words and Turing machines

We describe Turing machines, tilings and in#nite words as dynamical systems and analyze some of their dynamical properties. It is known that some of these systems do not always have periodic con#gurations; we prove that they always have quasi-periodic con#gurations and we quantify quasi-periodicity. We then study the decidability of dynamical properties for these systems. In analogy to Rice’s theorem for computable functions, we derive a theorem that characterizes dynamical system properties that are undecidable. As an illustration of this result, we prove that topological entropy is undecidable for Turing machines and for tilings. c © 2004 Elsevier B.V. All rights reserved.