Université Paris-Est Université Paris-Est - Marne-la-Vallée Université Paris-Est - Créteil Val-de-Marne Centre National de la Recherche Scientifique

Topological Hausdorff dimension

25/01/2012 - 11:00 - 12:00
Université Loránd Eötvös

This is a joint work with R. Balka and M. Elekes.

We introduce a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. We examine the basic properties of this new notion of dimension, compare it to other well-known notions, determine its value for some classical fractals such as the Sierpinski carpet, the von Koch snowflake curve, Kakeya sets, the trail of the Brownian motion, etc.

For a compact metric space $K$ let $\dim_H K$ and $\dim_{tH} K$ denote its Hausdorff and topological Hausdorff dimension, respectively. We show that if M denotes the limit set of Mandelbrot’s fractal percolation process then for every $d\in [0, 2)$ there exists a critical $p_c^{(d)}=p_c^{(d,n)}\in (0, 1)$ such that if $p < p_c^{(d)}$ then $\dim_{tH} M \le d$ almost surely, and if $p > p_c^{(d)}$ then $\dim_{tH} M > d$ almost surely (provided $M \neq\emptyset$).

We prove that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on $K$, namely $\sup\{\dim_H f^{−1} (y) : y ∈ \mathbb{R}\} = \dim_{tH} K − 1$ for the generic $f \in C(K)$. The supremum is actually attained, and there may only be a unique level set of maximal Hausdorff dimension.

We also show 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 $y\in f (K)$.

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 $y \in \text{int}\, f (K)$ 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 $K$.