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.

翻译：我们探究了动力系统的图灵完全性与拓扑熵之间的关系。首先证明了一类被称为“正则图灵机”（涵盖通用图灵机的典型实例）的自然图灵机均具有正拓扑熵。由此推导得出：任何采用连续编码模拟该类通用图灵机的图灵完全动力系统均呈现混沌特性。该结论适用于我们此前构造的圆盘保面积微分同胚与三维稳态欧拉流的图灵完全系统。本文以Ville Salo撰写的附录作结，该附录提出了一种构造非正则且拓扑熵为零的通用图灵机的新方法。