哈尔分解与埃利斯流的顺从性

中文摘要 AI 辅助

设\((X,G)\)为温顺流，\(K\)为其包络半群\(E(X,G)\)的埃利斯群。尽管\(K\)在其\(\tau\)-拓扑中是紧致豪斯多夫拓扑群，但\(K\)到\(E(X,G)\)的包含映射不一定是博雷尔的。我们证明，通过里斯 - 马尔可夫定理，\(K\)上的归一化哈尔测度确定了\(E(X,G)\)上的一个规范正则博雷尔概率测度\(\mu_K\)，称为其哈尔分解。我们的主要结构结果表明，对于每个温顺流，\((E(X,G),G)\)是顺从的当且仅当\((X,G)\)是遗传顺从的。对于温顺的遗传顺从流，当\(X\)是可度量的或\(G\)是可数的时，每个哈尔分解都是\(G\)-不变的。对于具有不变测度的可度量极小温顺流，每个哈尔分解在每个点的求值推送是唯一的不变测度。此外，可度量温顺流上的每个遍历不变测度都有最小支撑；因此，可度量温顺轨道上的每个遍历不变测度都是通过对合适的哈尔分解求值得到的。

英文摘要

Let $(X,G)$ be a tame flow and let $K$ be an Ellis group of its enveloping semigroup $E(X,G)$. Although $K$ is a compact Hausdorff topological group in its $τ$-topology, the inclusion $K\hookrightarrow E(X,G)$ need not be Borel. We show that normalized Haar measure on $K$ nevertheless determines, via the Riesz--Markov theorem, a canonical regular Borel probability measure $μ_K$ on $E(X,G)$, called its Haar decompression. Haar decompression gives an exact ergodicity description. For an arbitrary flow $(X,G)$, amenability of its Ellis flow is equivalent to hereditary amenability of all finite powers $X^n$. In the tame setting, $(E(X,G),G)$ is amenable if and only if $(X,G)$ is hereditarily amenable. Moreover, for the minimal left ideal $\mathcal{M}$ in $E(X,G)$ containing $K$, $μ_K$ is $G$-invariant if and only if $(\mathcal{M},G)$ is amenable; when this holds, $μ_K$ is the unique ergodic measure on $\mathcal{M}$ and $\mathcal{M}=\overline{K}$. Then, we conclude that the ergodic measures of $E(X,G)$ are precisely the Haar decompressions associated with amenable minimal left ideals. We also prove that every minimal tame amenable flow is uniquely ergodic and that every ergodic invariant measure on a tame flow has minimal support. Consequently, every ergodic measure on a tame ambit arises by evaluating a suitable Haar decompression.

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39 pages