The dimension of projections of fractal percolations

Abstract

\emph{Fractal percolation} or \emph{Mandelbrot percolation} is one of the most well studied families of random fractals. In this paper we study some of the geometric measure theoretical properties (dimension of projections and structure of slices) of these random sets. Although random, the geometry of those sets is quite regular. Our results imply that, denoting by $E\subset {\mathbb{R}}^2$ a typical realization of the fractal percolation on the plane, {itemize} If ${\dim }_{\mathrm{H}}E<1$ then for \textbf{all}lines $\ell$ the orthogonal projection $E_{\ell }$ of $E$ to $\ell$ has the same Hausdorff dimension as $E$, If ${\dim }_{\mathrm{H}}E>1$ then for any smooth real valued function $f$ which is strictly increasing in both coordinates, the image $f(E)$ contains an interval. {itemize} The second statement is quite interesting considering the fact that $E$ is almost surely a Cantor set (a {\it random dust}) for a large part of the parameter domain, see \cite{Chayes1988}. Finally, we solve a related problem about the existence of an interval in the algebraic sum of $d\ge 2$ one-dimensional fractal percolations.

Corrected version

Cited by in corpus (7)

• Exact dimensionality and projections of random self-similar measures and sets
• Distribution of random Cantor sets on Tubes
• Sets which are not tube null and intersection properties of random measures
• Almost sure convergence and second moments of geometric functionals of fractal percolation
• Dimension conservation for self-similar sets and fractal percolation
• Projections of random covering sets
• Projections of planar Mandelbrot measures