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 $\mathrm{Int}_\mathrm{top} (G,\mathcal{U}|Y)$ and average sample complexity $\mathrm{Asc}_\mathrm{top} (G,\mathcal{U}|Y)$. Given an invariant measure $\mu$ of $X$, we study the conditional local measure-theoretical intricacy $\mathrm{Int}_\mu^\pm(G,\mathcal{U}|Y)$ and average sample complexity $\mathrm{Asc}_\mu^\pm(G,\mathcal{U}|Y)$. For any F{\o}lner sequence $\{F_n\}_{n\in\mathbb{N}}$, we take $\{c^{F_n}_S\}_{S\subseteq F_n}$ to be the uniform system of coefficients. We establish the equivalence of $\mathrm{Asc}_\mu^-(G,\mathcal{U}|Y)$ and $\mathrm{Asc}_\mu^+(G,\mathcal{U}|Y)$ when $G=\mathbb{Z}$. Furthermore, we verified that $\mathrm{Asc}_\mu^-(G,\mathcal{U})$ is equal to $\mathrm{Asc}_\mu^+(G,\mathcal{U})$ in general case. Finally, we give a local variational principle of average sample complexity.