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 "branching Turing machines" (which includes most of the known examples of universal Turing machines) has positive topological entropy. Motivated by the recent construction of Turing complete Euler flows, we deduce that any Turing complete dynamics with a continuous encoding that simulates a universal branching machine is chaotic. On the other hand, we show that, unexpectedly, universal Turing machines with zero topological entropy (and even zero speed) can be constructed, unveiling the independence of chaos and universality at the symbolic level.

翻译：我们研究了动力系统的图灵完全性与拓扑熵之间的关系。首先证明了一类自然定义的图灵机（称为“分支图灵机”，包含大多数已知通用图灵机实例）具有正拓扑熵。受近年来图灵完全欧拉流构造的启发，我们得出推论：任何通过连续编码模拟通用分支机的图灵完全动力系统都是混沌的。另一方面，我们意外证明可以构造出拓扑熵为零（甚至速度为0）的通用图灵机，揭示了在符号层次上混沌性与通用性的独立性。