We begin development of a method for studying dynamical systems using concepts from computational complexity theory. We associate families of decision problems, called telic problems, to dynamical systems of a certain class. These decision problems formalize finite-time reachability questions for the dynamics with respect to natural coarse-grainings of state space. Our main result shows that complexity-theoretic lower bounds have dynamical consequences: if a system admits a telic problem for which every decider runs in time $2^{Ω(n)}$, then it must have positive topological entropy. This result and others lead to methods for classifying dynamical systems through proving bounds on the runtime of algorithms solving their associated telic problems, or by constructing polynomial-time reductions between telic problems coming from distinct dynamical systems.

翻译：我们开始开发一种利用计算复杂性理论概念研究动力学系统的方法。我们将称为目的问题的决策问题族与特定类别的动力学系统相关联。这些决策问题形式化了动力学关于状态空间自然粗粒化的有限时间可达性问题。我们的主要结果表明，复杂性理论下界具有动力学意义：如果一个系统存在一个目的问题，使得所有判定器都需要 $2^{Ω(n)}$ 时间运行，则该系统必须具有正拓扑熵。这一结果及其他发现引出了通过证明求解相关目的问题的算法运行时间下界，或通过构建来自不同动力学系统的目的问题之间的多项式时间归约，来对动力学系统进行分类的方法。