Inductive topological Hausdorff dimensions and fibers of generic continuous functions

In an earlier paper Z. Buczolich, M. Elekes and the author introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. They proved that it is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space $K$. The goal of this paper is to determine the Hausdorff dimension of the fibers of the generic continuous function from $K$ to $\RR^n$. In order to do so, we define the $n$th inductive topological Hausdorff dimension, $\dim_{t^nH} K$. Let $\dim_H K$, $\dim_t K$ and $C_n(K)$ denote the Hausdorff and topological dimension of $K$ and the Banach space of the continuous functions from $K$ to $\RR^n$. The quintessence of our main theorem is that $\sup_{y\in \RR^n} \dim_{H}f^{-1}(y) = \dim_{t^nH} K -n$ for the generic $f \in C_n(K)$, provided that $\dim_t K\geq n$, otherwise every fiber is finite. The most interesting ingredient of the proof is that we modify and use topological methods coming from dimension theory. We show that the supremum is actually attained on the left hand side of the above equation. We characterize those compact metric spaces for which $\dim_{H} f^{-1}(y)=\dim_{t^nH}K-n$ for the generic $f\in C_n(K)$ and the generic $y\in f(K)$. We also generalize a result of B. Kirchheim by showing that if $K$ is self-similar and $\dim_t K\geq n$ then $\dim_{H} f^{-1}(y)=\dim_{t^nH}K-n$ for the generic $f\in C_n(K)$ for every $y\in \inter f(K)$.