论文标题
关于几乎有限动作的扩展的嵌入到立方移动中
On embeddings of extensions of almost finite actions into cubical shifts
论文作者
论文摘要
对于可数的Amenable Group $ G $和固定尺寸$ M \ GEQ 1 $，我们调查何时可以将$ g $ -space $ x $嵌入到$ M $ $二维立方移位$（[0,1]^m）^g $中。从Matui和Kerr的意义上讲，我们将注意力集中在完全断开的空间$ y $上的几乎有限的$ g $ Action的系统上。我们表明，如果这样的$ g $ - 空间$ x $的尺寸小于$ m/2 $，则$ x $嵌入$（m+1）$ - 尺寸 - 尺寸立方体移位。如果假定杰出因子$ g $ -space $ y $是对有限类型的子迁移，则可以将其改进到嵌入到$ m $二维的立方体位置中。 Gutman-tsukamoto应该将此结果视为定理的概括，以$ G = \ Mathbb Z $对所有不友善组的动作，这代表了支持Lindenstrauss-Tsukamoto猜想的第一个结果，用于$ G = \ Mathbb {z}^k $以外的其他组的行动。
For a countable amenable group $G$ and a fixed dimension $m\geq 1$, we investigate when it is possible to embed a $G$-space $X$ into the $m$-dimensional cubical shift $([0,1]^m)^G$. We focus our attention on systems that arise as an extension of an almost finite $G$-action on a totally disconnected space $Y$, in the sense of Matui and Kerr. We show that if such a $G$-space $X$ has mean dimension less than $m/2$, then $X$ embeds into the $(m+1)$-dimensional cubical shift. If the distinguished factor $G$-space $Y$ is assumed to be a subshift of finite type, then this can be improved to an embedding into the $m$-dimensional cubical shift. This result ought to be viewed as the generalization of a theorem by Gutman-Tsukamoto for $G=\mathbb Z$ to actions of all amenable groups, and represents the first result supporting the Lindenstrauss-Tsukamoto conjecture for actions of groups other than $G=\mathbb{Z}^k$.
