Veech's Theorem of $G$ acting freely on $G^{\textrm{LUC}}$ and Structure Theorem of a.a. flows
| Autor: | Dai, Xiongping, Liang, Hailan, Yin, Zhengyu |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Veech's Theorem claims that if $G$ is a locally compact\,(LC) Hausdorff topological group, then it may act freely on $G^{\textrm{LUC}}$. We prove Veech's Theorem for $G$ being only locally quasi-totally bounded, not necessarily LC. And we show that the universal a.a. flow is the maximal almost 1-1 extension of the universal minimal a.p. flow and is unique up to almost 1-1 extensions. In particular, every endomorphism of Veech's hull flow induced by an a.a. function is almost 1-1; for $G=\mathbb{Z}$ or $\mathbb{R}$, $G$ acts freely on its canonical universal a.a. space. Finally, we characterize Bochner a.a. functions on a LC group $G$ in terms of Bohr a.a. function on $G$ (due to Veech 1965 for the special case that $G$ is abelian, LC, $\sigma$-compact, and first countable). Comment: 54 pages |
| Databáze: | arXiv |
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