Topological mixing notions on Turing machine dynamical systems

Over the past few decades, Turing machines have been studied as dynamical systems, with the focus being on their behavior rather than their results. Noteworthy results concerning topological and dynamical properties, such as the existence and undecidability of topological transitivity in TMH and topological minimality in TMT, were established. Both properties are related to reaching finite windows from some or any possible configuration. Nonetheless, both properties exhibit no restriction over the time a machine takes to reach these finite windows. In this article, we focus on the mixing notions: weak mixing, total transitivity and topological mixing. These properties are related to a time window or gap where finite configurations must reach one another. In this article, we analyze the SMART machine to prove that its TMT dynamical model is topologically weak mixing (and therefore totally transitive) and that all mixing notions are undecidable. (c) 2022 Elsevier Inc. All rights reserved.