In this paper, we study a dichotomy of higher-order repeated time averages of a compact dynamical system acting on an infinite dimensional Banach space. We also discuss one feature so-called uniformly historic behaviour of a compact dynamical system which causes the non-existence of time averages. In its own turn, this feature will eventually cause the non-existence of higher-order repeated time averages. The best example for a compact dynamical system having uniformly historic behaviour is a discrete-time Kolmogorov system of predator–prey interactions. As an application, we also provide some examples of three-species predator–prey interactions in which uniformly historic behaviour can be observed.

在本文中，我们研究了作用于无限维 Banach 空间的紧凑动力系统的高阶重复时间平均值的二分法。我们还讨论了一个称为紧凑动力系统的一致历史行为的特征，它导致时间平均值不存在。反过来，这个特征最终会导致高阶重复时间平均值的不存在。具有一致历史行为的紧凑动力系统的最佳示例是捕食者-猎物相互作用的离散时间 Kolmogorov 系统。作为一个应用，我们还提供了一些三物种捕食者-猎物相互作用的例子，在这些例子中可以观察到一致的历史行为。