Topological Hausdorff dimension and level sets of generic continuous functions on fractals

In an earlier paper we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space K let dimHK and dimtHK denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on K, namely sup{dimHf-1(y):y∈R}=dimtHK-1 for the generic f∈C(K), provided that K is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if K is not totally disconnected and sufficiently homogeneous then dimHf−1(y)=dimtHK−1 for the generic f∈C(K) and the generic y∈f(K). The most important goal of this paper is to make these theorems more precise.