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

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

2011-08-29

arxiv.org/abs/1108.5578v1

Mathematics

Classical Analysis and ODEs

24 pages

Scientific paper

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 $\sup\{\dim_{H}f^{-1}(y) : y \in \Reals \} = \dim_{tH} K - 1$ for the generic $f \in C(K)$. We also proved that if $K$ is 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, 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 $y\in \interior f(K)$ 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 $K$.

Affiliated with

Also associated with

No associations

Rating

Topological Hausdorff dimension and level sets of generic continuous functions on fractals does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Topological Hausdorff dimension and level sets of generic continuous functions on fractals, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Topological Hausdorff dimension and level sets of generic continuous functions on fractals will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-126482