一般连续映射纤维的维数
Dimensions of fibers of generic continuous maps
摘要：
在较早的文件Buczolich，Elekes和作者描述了Hausdorff维数的水平集的一般实值连续函数（在意义上的Baire类别）定义在一个紧凑的度量空间$K$。后来，作者将$K$映射理论推广到$\mathbb{R}^n$。本文的主要目的是推广拓扑维数和packing维数的相关结果。设$K$是一个紧度量空间，用$C(K,\mathbb{R}^n)$表示从$K$到$\mathbb{R}^n$的具有最大范数的连续映射的集合.令$\dim_{*}$为拓扑维数$\dim_T$、Hausdorff维数$\dim_H$或填充维数$\dim_P$之一。定义了$$d_{*}^n(K)=\inf\{\dim_{*}(K\setminus F): F\subset K \textrm{ is $σ$-compact with } \dim_T F<n\}.$$我们证明了$d^n_{*}(K)$是描述一般连续映射$f\in C(K,\mathbb{R}^n)$纤维维数的正确概念。特别是，我们证明了$\sup\{\dim_{*}f^{-1}(y): y\in \mathbb{R}^n\} =d^n_{*}(K)$，如果$\dim_T K\geq n$，否则每个纤维是有限的。要证明上述关于填充维数的定理，需要全新的思想。此外，我们证明了上确界是在上述方程的左侧得到的。假定为$\dim_T K\geq n$。如果$K$足够均匀，那么我们可以说得更多。例如，我们证明了，$\dim_{*}f^{-1}(y)=d^n_{*}(K)$为一般$f\in C(K,\mathbb{R}^n)$为所有$y\in \textrm{int} f(K)$当且仅当$d^n_{*}(U)=d^n_{*}(K)$或$\dim_T U<n$为所有开放集$U\subset K$。这是新的，即使$n=1$和$\dim_{*}=\dim_H$。已知对于通用$f\in C(K,\mathbb{R}^n)$，$f(K)$的内部不为空。我们通过证明通用$f\in C(K,\mathbb{R}^n)$的$\dim_T \partial f(K)=\dim_H \partial f(K)=n-1$来增强上述表征。
英文摘要：
In an earlier paper Buczolich, Elekes and the author described the Hausdorff dimension of the level sets of a generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space $K$. Later on, the author extended the theory for maps from $K$ to $\mathbb{R}^n$. The main goal of this paper is to generalize the relevant results for topological and packing dimensions. Let $K$ be a compact metric space and let us denote by $C(K,\mathbb{R}^n)$ the set of continuous maps from $K$ to $\mathbb{R}^n$ endowed with the maximum norm. Let $\dim_{*}$ be one of the topological dimension $\dim_T$, the Hausdorff dimension $\dim_H$, or the packing dimension $\dim_P$. Define $$d_{*}^n(K)=\inf\{\dim_{*}(K\setminus F): F\subset K \textrm{ is $σ$-compact with } \dim_T F<n\}.$$ We prove that $d^n_{*}(K)$ is the right notion to describe the dimensions of the fibers of a generic continuous map $f\in C(K,\mathbb{R}^n)$. In particular, we show that $\sup\{\dim_{*}f^{-1}(y): y\in \mathbb{R}^n\} =d^n_{*}(K)$ provided that $\dim_T K\geq n$, otherwise every fiber is finite. Proving the above theorem for packing dimension requires entirely new ideas. Moreover, we show that the supremum is attained on the left hand side of the above equation. Assume $\dim_T K\geq n$. If $K$ is sufficiently homogeneous, then we can say much more. For example, we prove that $\dim_{*}f^{-1}(y)=d^n_{*}(K)$ for a generic $f\in C(K,\mathbb{R}^n)$ for all $y\in \textrm{int} f(K)$ if and only if $d^n_{*}(U)=d^n_{*}(K)$ or $\dim_T U<n$ for all open sets $U\subset K$. This is new even if $n=1$ and $\dim_{*}=\dim_H$. It is known that for a generic $f\in C(K,\mathbb{R}^n)$ the interior of $f(K)$ is not empty. We augment the above characterization by showing that $\dim_T \partial f(K)=\dim_H \partial f(K)=n-1$ for a generic $f\in C(K,\mathbb{R}^n)$.
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