Local intricacy and average sample complexity for amenable group actions

Let $(X,G)$, $(Y,G)$ be two $G$-systems, where $G$ is an infinite countable discrete amenable group and $X$, $Y$ are compact metric spaces. Suppose that $\mathcal{U}$ is a cover of $X$. We first introduce the conditional local topological intricacy ${\mathrm{I}\mathrm{n}\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{p}}(G,\mathcal{U}\mathrm{\mid }Y)$ and average sample complexity ${\mathrm{A}\mathrm{s}\mathrm{c}}_{\mathrm{t}\mathrm{o}\mathrm{p}}(G,\mathcal{U}\mathrm{\mid }Y)$. Given an invariant measure $\mu$ of $X$, we study the conditional local measure-theoretical intricacy ${\mathrm{I}\mathrm{n}\mathrm{t}}_{\mu }^{\pm }(G,\mathcal{U}\mathrm{\mid }Y)$ and average sample complexity ${\mathrm{A}\mathrm{s}\mathrm{c}}_{\mu }^{\pm }(G,\mathcal{U}\mathrm{\mid }Y)$. For any Følner sequence $\{F_n{\}}_{n\in \mathbb{N}}$, we take $\{c_S^{F_n}{\}}_{S\subseteq F_n}$ to be the uniform system of coefficients. We establish the equivalence of ${\mathrm{A}\mathrm{s}\mathrm{c}}_{\mu }^{-}(G,\mathcal{U}\mathrm{\mid }Y)$ and ${\mathrm{A}\mathrm{s}\mathrm{c}}_{\mu }^{+}(G,\mathcal{U}\mathrm{\mid }Y)$ when $G=\mathbb{Z}$. Furthermore, we verified that ${\mathrm{A}\mathrm{s}\mathrm{c}}_{\mu }^{-}(G,\mathcal{U})$ is equal to ${\mathrm{A}\mathrm{s}\mathrm{c}}_{\mu }^{+}(G,\mathcal{U})$ in general case. Finally, we give a local variational principle of average sample complexity.