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：我们研究在有限但任意小预设误差下严格描述拓扑动力系统渐近行为的计算问题。更精确地说，我们考虑典型轨道的极限集，既作为空间对象（吸引子集）又作为统计分布（物理测度），并证明以任意精度计算这些对象描述的计算资源上界。我们还研究不同动力学约束如何影响这些上界，并提供多个例子表明这些上界在一般情况下是紧的。特别地，我们构造了一个可计算的区间映射，该映射具有唯一传递吸引子且支持唯一物理测度，该吸引子具有康托尔集结构，使得吸引子和测度均不可计算。