We study the computational problem of rigorously describing the asymptotic behaviour of topological dynamical systems up to a finite but arbitrarily small pre-specified error. More precisely, we consider the limit set of a typical orbit, both as a spatial object (attractor set) and as a statistical distribution (physical measure), and prove upper bounds on the computational resources of computing descriptions of these objects with arbitrary accuracy. We also study how these bounds are affected by different dynamical constrains and provide several examples showing that our bounds are sharp in general. In particular, we exhibit a computable interval map having a unique transitive attractor with Cantor set structure supporting a unique physical measure such that both the attractor and the measure are non computable.

翻译：我们研究拓扑动力系统渐近行为的计算问题，要求以有限但可任意小的预先指定误差进行严格描述。具体而言，我们同时考虑典型轨道的极限集作为空间对象（吸引子集）和统计分布（物理测度），并证明了以任意精度计算这些对象描述所需计算资源的上界。我们还研究了不同动力学约束如何影响这些界，并通过若干示例证明我们的界在一般情况下是最优的。特别地，我们构造了一个可计算的区间映射，其具有唯一的传递吸引子，该吸引子具有康托尔集结构并承载唯一的物理测度，但该吸引子与测度均不可计算。