平均维数与Jaworski型定理
Mean Dimension & Jaworski-type Theorems
摘要：
根据著名的Jaworski定理，有限维非周期动力系统$(X,T)$嵌入$1$ -维立方位移$([0,1]^{\mathbb{Z}},shift)$中.如果$X$承认周期点（仍然假设$\dim(X)<\infty$），那么我们在本文中表明，周期维度$perdim(X,T)<\frac{d}{2}$意味着$(X,T)$嵌入在$d$ -维立方移位$(([0,1]^{d})^{\mathbb{Z}},shift)$。这验证了Lindenstrauss和Tsukamoto对有限维系统的一个猜想。对于无穷维动力系统，在相同的周期维数假设下，周期点集可以等变地浸入$(([0,1]^{d})^{\mathbb{Z}},shift)$中。此外，我们引入了一般拓扑动力系统的标记的概念，并使用推广版本的Bonatti-Crockier塔定理，以显示一个非周期的有限维系统，其平均尺寸服从$mdim(X,T)<\frac{d}{16}$的扩展$(X,T)$嵌入在$(d+1)$ -立方移位$(([0,1]^{d+1})^{\mathbb{Z}},shift)$。
英文摘要：
According to the celebrated Jaworski Theorem, a finite dimensional aperiodic dynamical system $(X,T)$ embeds in the $1$-dimensional cubical shift $([0,1]^{\mathbb{Z}},shift)$. If $X$ admits periodic points (still assuming $\dim(X)<\infty$) then we show in this paper that periodic dimension $perdim(X,T)<\frac{d}{2}$ implies that $(X,T)$ embeds in the $d$-dimensional cubical shift $(([0,1]^{d})^{\mathbb{Z}},shift)$. This verifies a conjecture by Lindenstrauss and Tsukamoto for finite dimensional systems. Moreover for an infinite dimensional dynamical system, with the same periodic dimension assumption, the set of periodic points can be equivariantly immersed in $(([0,1]^{d})^{\mathbb{Z}},shift)$. Furthermore we introduce a notion of markers for general topological dynamical systems, and use a generalized version of the Bonatti-Crovisier tower theorem, to show that an extension $(X,T)$ of an aperiodic finite-dimensional system whose mean dimension obeys $mdim(X,T)<\frac{d}{16}$ embeds in the $(d+1)$-cubical shift $(([0,1]^{d+1})^{\mathbb{Z}},shift)$.
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