Inductive topological Hausdorff dimensions and fibers of generic continuous functions

Abstract

In an earlier paper Buczolich, 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 \(\mathbb {R}^n\). 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 \(\mathbb {R}^n\). We show that \(\sup _{y\in \mathbb {R}^n} \dim _{H}f^{-1}(y) = \dim _{t^nH} K -n\) for the generic \(f \in C_n(K)\), provided that \(\dim _t K\ge n\), otherwise every fiber is finite. In order to prove the above theorem we give some equivalent definitions for the inductive topological Hausdorff dimensions, which can be interesting in their own right. Here we use techniques coming from the theory of topological dimension. We show that the supremum is actually attained on the left hand side of the above equation. We characterize those compact metric spaces \(K\) 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 Kirchheim by showing that if \(K\) is self-similar and \(\dim _t K\ge n\) then \(\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n\) for the generic \(f\in C_n(K)\) for every \(y\in {{\mathrm{int}}}f(K)\).