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

Research paper by **Richard Balka, Zoltan Buczolich, Marton Elekes**

Indexed on: **04 Aug '12**Published on: **04 Aug '12**Published in: **Mathematics - Classical Analysis and ODEs**

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#### Abstract

In an earlier paper (arxiv:1108.4292) we introduced a new concept of
dimension for metric spaces, the so called topological Hausdorff dimension. For
a compact metric space $K$ let $\dim_{H}K$ and $\dim_{tH} K$ 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 $\sup{\dim_{H}f^{-1}(y) : y \in
\mathbb{R}} = \dim_{tH} K - 1$ for the generic $f \in C(K)$, 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 $\dim_{H}f^{-1}(y) = \dim_{tH} K - 1$ for the
generic $f \in C(K)$ and the generic $y \in f(K)$. The most important goal of
this paper is to make these theorems more precise.
As for the first result, we prove that the supremum is actually attained on
the left hand side of the first equation above, and also show that there may
only be a unique level set of maximal Hausdorff dimension.
As for the second result, we characterize those compact metric spaces for
which for the generic $f\in C(K)$ and the generic $y\in f(K)$ we have $\dim_{H}
f^{-1}(y)=\dim_{tH}K-1$. We also generalize a result of B. Kirchheim by showing
that if $K$ is self-similar then for the generic $f\in C(K)$ for every $y\in
\inter f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$.
Finally, we prove that the graph of the generic $f\in C(K)$ has the same
Hausdorff and topological Hausdorff dimension as $K$.