The pressure of intricacy and average sample complexity for amenable group actions

Let (X, G) be a G-system, where G is an infinite countable discrete amenable group and X is a compact metric space with a metric d. In this paper, we study the topological pressure of intricacy and average sample complexity for amenable group actions. We show that the topological pressure of strong sub-additive potential is equal to the pressure of intricacy and average sample complexity as taking supremum over open covers of X. We establish the definition of the pressure of intricacy and average sample complexity by using spanning sets and separated sets. In the last part, we show that the variational principle for pressure of intricacy and average sample complexity.