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 $K$. 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 $$d_{*}^n(K)=\inf\{\dim_{*}(K\setminus F): F\subset K \textrm{ is $\sigma$-compact with } \dim_T F<n\}.$$ 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 $f\in C(K,\mathbb{R}^n)$.