In this paper, we consider chaos of a finite family of continuous functions. As a measure of chaos, we use three types of entropies defined for that family. The first type of entropy is connected with the entropy of semigroups while the second and the third type concern entropy of nonautonomous dynamical systems. The main aim of the paper is to analyze the local aspects related to these concepts. To this end, we consider three types of points accumulating entropy and we investigate their existence, differences between them, and the possibility of disruptions.
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