We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \textrm{C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

中文翻译：

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局部路径连通紧致空间上流动的等连续因子

我们考虑一个局部路径连接的紧凑度量空间ķ具有有限的第一个 Betti 数$\textrm {b}_1(K)$和一个流程$(K, G)$在ķ这样G是阿贝尔和所有G- 不变函数$f\,{\in}\,\textrm{C}(K)$是恒定的。我们证明了流动的每一个等连续因子$(K, G)$同构于维数小于的紧阿贝尔李群上的流${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$. 为此，我们使用并提供 Hauser 和 Jäger 的定理 2.12 的新证明[最大等连续因子的单调性和对环流的应用。过程。阿米尔。数学。社会党。147(2019), 4539–4554]，其中指出，对于局部连通紧凑空间上的流，最大等连续因子上的商映射为单调，即具有连接的纤维。我们的替代证明是对商图单调性的新表征的简单结果$p\冒号 K\to L$在局部连通的紧致空间之间ķ和大号我们通过表征ķ就 Banach 晶格而言$\textrm {C}(K)$.