论文标题
具有正平均值的连续图的通用性
Genericity of continuous maps with positive metric mean dimension
论文作者
论文摘要
格罗莫夫（M. Gromov）在1990年代后期引入了连续地图的平均维度，这在拓扑结合下是不变的。另一方面，Lindenstrauss和Weiss在2000年引入了动态系统的度量平均维度概念，这完善了具有无限拓扑熵的动力学系统的拓扑熵。在本文中，我们将显示，如果$ n $是$ n $尺寸紧凑的riemannian歧管，那么，对于[0，n] $中的任何$ a \，由$ c^{0}（n）$的连续映射组成的集合组成的集合，对于$ c^{0}（n）$，对于$ a = n $均为残留。此外，我们证明了一些与具有正平均值的cantor集合和具有正平均尺寸的乘积空间上的连续图的存在和密度有关的结果。
M. Gromov introduced the mean dimension for a continuous map in the late 1990's, which is an invariant under topological conjugacy. On the other hand, the notion of metric mean dimension for a dynamical system was introduced by Lindenstrauss and Weiss in 2000 and this refines the topological entropy for dynamical systems with infinite topological entropy. In this paper we will show if $N$ is a $n$ dimensional compact riemannian manifold then, for any $a\in [0,n]$, the set consisting of continuous maps with metric mean dimension equal to $a$ is dense in $C^{0}(N)$ and for $a=n$ this set is residual. Furthermore, we prove some results related to existence and density of continuous maps on Cantor sets with positive metric mean dimension and on product spaces with positive mean dimension.
