Uniformly historic behaviour in compact dynamical systems

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.

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