Dimensions of fibers of generic continuous maps

Abstract

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 by introducing the notion of topological Hausdorff dimension. 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 and to obtain new results for sufficiently homogeneous spaces K even in the case case of Hausdorff dimension. 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

$$\begin{aligned} d_{*}^n(K)=\inf \left\{ \dim _{*}(K{\setminus } F): F\subset K \text { is } \sigma \text {-compact with } \dim _T F<n\right\} . \end{aligned}$$

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 $\dim _T K\ge n$, 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 $\dim _T K\ge n$. 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 $y\in {{\mathrm{int}}}f(K)$ 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 $f\in C(K,{\mathbb {R}}^n)$. In particular, almost every point of f(K) is an interior point. In order to obtain more precise results, we use the concept of generalized Hausdorff and packing measures, too.