Local intricacy and average sample complexity for amenable group actions
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918147876978688 |
|---|---|
| author | Huang, J. Xiao, Z. |
| author_facet | Huang, J. Xiao, Z. |
| contents | 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 $μ$ of $X$, we study the conditional local measure-theoretical intricacy $\mathrm{Int}_μ^\pm(G,\mathcal{U}|Y)$ and average sample complexity $\mathrm{Asc}_μ^\pm(G,\mathcal{U}|Y)$. For any Følner 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 $\mathrm{Asc}_μ^-(G,\mathcal{U}|Y)$ and $\mathrm{Asc}_μ^+(G,\mathcal{U}|Y)$ when $G=\mathbb{Z}$. Furthermore, we verified that $\mathrm{Asc}_μ^-(G,\mathcal{U})$ is equal to $\mathrm{Asc}_μ^+(G,\mathcal{U})$ in general case. Finally, we give a local variational principle of average sample complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_20738 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local intricacy and average sample complexity for amenable group actions Huang, J. Xiao, Z. Dynamical Systems 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 $μ$ of $X$, we study the conditional local measure-theoretical intricacy $\mathrm{Int}_μ^\pm(G,\mathcal{U}|Y)$ and average sample complexity $\mathrm{Asc}_μ^\pm(G,\mathcal{U}|Y)$. For any Følner 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 $\mathrm{Asc}_μ^-(G,\mathcal{U}|Y)$ and $\mathrm{Asc}_μ^+(G,\mathcal{U}|Y)$ when $G=\mathbb{Z}$. Furthermore, we verified that $\mathrm{Asc}_μ^-(G,\mathcal{U})$ is equal to $\mathrm{Asc}_μ^+(G,\mathcal{U})$ in general case. Finally, we give a local variational principle of average sample complexity. |
| title | Local intricacy and average sample complexity for amenable group actions |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2509.20738 |