拓扑动力学中的嵌入问题与Takens定理
The embedding problem in topological dynamics and Takens' theorem
摘要：
证明了每个平均维数小于$D/2$的$\mathbb{Z}^{k}$ -作用$(X,\mathbb{Z}^{k},T)$嵌入$(([0,1]^{(L+1)D})^{\mathbb{Z}^{k}}\times Y,σ\times S)$，其中$D\in\mathbb{N}$，$L\in\mathbb{N}\cup\{0\}$和$σ$是Hilbert立方体$([0,1]^{(L+1)D})^{\mathbb{Z}^{k}}$上的移位;特别地，当$(Y,\mathbb{Z}^{k},S)$是$k$ -环面上的无理$\mathbb{Z}^{k}$ -旋转时，$(X,\mathbb{Z}^{k},T)$嵌入在$(([0,1]^{2^kD+1})^{\mathbb{Z}^k},σ)$中，该结果与第一作者Lindenstrauss和Tsukamoto的先前结果进行了比较。此外，我们对$\mathbb{Z}$ -作用给出了Takens嵌入定理的一个完整而详细的证明，并对$\mathbb{Z}^{k}$ -作用推出了类似的结果.最后，我们证明了Lindenstrauss-Tsukamoto猜想对$\mathbb{Z}$ -作用的一般成立，讨论了前两位作者和Tsukamoto在即将发表的论文中对$\mathbb{Z}^{k}$ -作用的一个类似猜想，并对有限维空间上的$\mathbb{Z}^{k}$ -作用进行了验证.
英文摘要：
We prove that every $\mathbb{Z}^{k}$-action $(X,\mathbb{Z}^{k},T)$ of mean dimension less than $D/2$ admitting a factor $(Y,\mathbb{Z}^{k},S)$ of Rokhlin dimension not greater than $L$ embeds in $(([0,1]^{(L+1)D})^{\mathbb{Z}^{k}}\times Y,σ\times S)$, where $D\in\mathbb{N}$, $L\in\mathbb{N}\cup\{0\}$ and $σ$ is the shift on the Hilbert cube $([0,1]^{(L+1)D})^{\mathbb{Z}^{k}}$; in particular, when $(Y,\mathbb{Z}^{k},S)$ is an irrational $\mathbb{Z}^{k}$-rotation on the $k$-torus, $(X,\mathbb{Z}^{k},T)$ embeds in $(([0,1]^{2^kD+1})^{\mathbb{Z}^k},σ)$, which is compared to a previous result by the first named author, Lindenstrauss and Tsukamoto. Moreover, we give a complete and detailed proof of Takens' embedding theorem with a continuous observable for $\mathbb{Z}$-actions and deduce the analogous result for $\mathbb{Z}^{k}$-actions. Lastly, we show that the Lindenstrauss--Tsukamoto conjecture for $\mathbb{Z}$-actions holds generically, discuss an analogous conjecture for $\mathbb{Z}^{k}$-actions appearing in a forthcoming paper by the first two authors and Tsukamoto and verify it for $\mathbb{Z}^{k}$-actions on finite dimensional spaces.
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