Dynamical intricacy and average sample complexity of amenable group actions

In 2018, Petersen and Wilson introduced the notion of dynamical intricacy and average sample complexity for dynamical systems of ℤ-action, based on the past works on the notion of intricacy in the research of brain network and probability theory. If one wants to take into account underlying system geometry in applications, more general group actions may need to be taken into consideration. In this paper, we consider this notion in the case of amenable group actions. We show that many basic properties in the ℤ-action case remain true. We also show that their suprema over covers or partitions are equal to the amenable topological entropy and the measure entropy, using the quasitiling technique in the theory of the amenable group.